Therapy prediction and optimization of serum potassium for renal failure blood therapy, especially home hemodialysis

ABSTRACT

A method of predicting serum potassium concentrations in a patient during hemodialysis includes measuring serum potassium concentrations of the patient over a hemodialysis treatment session time and an ultrafiltration rate calculated by a difference between pre- and post-dialytic body weight of the patient during an initial hemodialysis treatment session divided by a total treatment time of the treatment session and estimating a potassium mobilization clearance and a pre-dialysis distribution volume of potassium for the patient. Serum potassium concentrations of the patient can then be predicted at any time during any hemodialysis treatment session with the estimated potassium mobilization clearance and pre-dialysis distribution volume of potassium of the patient.

PRIORITY CLAIM

This application claims priority to and the benefit as acontinuation-in-part application of U.S. patent application Ser. No.13/088,044, entitled, “Therapy Prediction and Optimization for RenalFailure Blood Therapy, Especially in Home Hemodialysis”, filed Apr. 15,2011, which claims priority to and the benefit of U.S. ProvisionalPatent Application Ser. No. 61/438,002 filed on Jan. 31, 2011, U.S.Provisional Patent Application Ser. No. 61/388,315 filed on Sep. 30,2010 and U.S. Provisional Patent Application Ser. No. 61/325,113 filedon Apr. 16, 2010, the entire disclosures of each of which are herebyincorporated by reference and relied upon.

BACKGROUND

The present disclosure generally relates to dialysis systems. Morespecifically, the present disclosure relates to therapy prediction andoptimization systems and methods for hemodialysis, especially homehemodialysis (“HHD”).

Hemodialysis (“HD”) in general uses diffusion to remove waste productsfrom a patient's blood. A diffusive gradient that occurs across thesemi-permeable dialysis membrane between the blood and an electrolytesolution called dialysate causes diffusion. Hemofiltration (“HF”) is analternative renal replacement therapy that relies on a convectivetransport of toxins from the patient's blood. This therapy isaccomplished by adding substitution or replacement fluid to theextracorporeal circuit during treatment (typically ten to ninety litersof such fluid). That substitution fluid and the fluid accumulated by thepatient in between treatments is ultrafiltered over the course of the HFtreatment, providing a convective transport mechanism that isparticularly beneficial in removing middle and large molecules (inhemodialysis there is a small amount of waste removed along with thefluid gained between dialysis sessions, however, the solute drag fromthe removal of that ultrafiltrate is not enough to provide convectiveclearance).

Hemodiafiltration (“HDF”) is a treatment modality that combinesconvective and diffusive clearances. HDF uses dialysate to flow througha dialyzer, similar to standard hemodialysis, providing diffusiveclearance. In addition, substitution solution is provided directly tothe extracorporeal circuit, providing convective clearance.

Home hemodialysis (“HHD”) has to date had limited acceptance even thoughthe clinical outcomes of this modality are more attractive thanconventional hemodialysis. There are benefits to daily hemodialysistreatments versus bi- or tri-weekly visits to a treatment center. Incertain instances, a patient receiving more frequent treatments removesmore toxins and waste products than a patient receiving less frequentbut perhaps longer treatments.

In any of the blood therapy treatments listed above, there is an artthat goes along with the science. Different patients will responddifferently to the same therapy. In centers where most dialysis takesplace, a patient's therapy is honed over time with the aid of staffclinicians or nurses. With home therapy, the patient will visit aclinician's or doctor's office on a regular, e.g., monthly basis, butwill not typically have a nurse or clinician at home to help optimizethe therapy. For this reason, a mechanism to aid in optimizing ahemodialysis or other blood therapy treatment early on after beginningdialysis is desirable.

Home therapy or HHD also provides the patient with therapy options thatthe in-center patient does not have. For example, HHD can performtherapy at night if desired, using a single or double needle modality.Any therapy, including a nighttime therapy can be performed over anamount of time that the patient can elect. Because the patient does nothave to commute to a center, the patient can perform therapy on daysthat are convenient for the patient, e.g., weekend days. Similarly, thepatient can choose a therapy frequency, or number of therapies per week,that is most convenient and/or most effective. With the addedflexibility comes questions, for example, the patient may wonder whetherit is better to run six therapies a week at 2.5 hours per therapy orfive therapies a week at three hours per therapy. For this additionalreason, not only is a way to help optimize a hemodialysis or other bloodtherapy treatment upfront desirable, it is also desirable to know whatwill happen when therapy parameters for the optimized therapy arechanged.

Optimizing hemodialysis therapies for a hemodialysis patient can also bedone by knowing the serum phosphorus levels of the hemodialysis patientduring and outside of a hemodialysis treatment session. However, serumphosphorus levels will vary depending on the type of hemodialysispatient and the characteristics of the hemodialysis treatment sessions.

Plasma or serum (the two terms will be used interchangeably) phosphoruskinetics during HD treatments cannot be explained by conventional one ortwo compartment models. Previous approaches have been limited byassuming that the distribution of phosphorus is confined to classicalintracellular and extracellular fluid compartments. More accuratekinetic models able to describe phosphorus kinetics during HD treatmentsand the post-dialytic rebound period during short HD (“SHD”) andconventional HD (“CHD”) treatment sessions can be used to predict steadystate, pre-dialysis serum phosphorus levels in patients treated with HDtherapies. This information can be useful in determining optimaltreatment regimens for hemodialysis patients.

Hemodialysis therapies can also be optimized by controlling serumpotassium levels of a hemodialysis patient during and outside of ahemodialysis treatment session. Plasma or serum potassium kineticsduring HD treatments cannot be explained by conventional one or twocompartment models. The kinetics of potassium during HD therapies cannotbe described using a conventional one-compartment model because theinterdialytic decreases in serum potassium concentration are differentthan those for urea, and there is a substantial post-dialysis rebound ofpotassium concentration. The kinetics of potassium during HD therapieshave been previously described using a two-compartment model assumingthat the distribution of potassium is confined to classic intracellularand extracellular fluid compartments with both active and passivetransport of potassium between the compartments, but suchtwo-compartment models are complex and require a relatively large numberof parameters to describe potassium kinetics. More simple and accuratepotassium kinetic models can therefore be useful in determining optimaltreatment regimens for hemodialysis patients.

SUMMARY

The present disclosure sets forth three primary embodiments. In a firstprimary embodiment, under Roman Numeral I below, the present disclosuresets forth systems and methods for a renal failure blood therapy, suchas, hemodialysis, hemofiltration, hemodiafiltration, and in particularfor home hemodialysis (“HHD”). The systems and methods of the presentdisclosure have in one embodiment three primary components, which can bestored on one or more computer, such as the computer for a doctor,clinician or nurse (referred to herein collectively as “doctor” unlessotherwise specified). The doctor's computer, can be in data-networkedcommunication with the renal failure therapy machine, e.g., via aninternet, a local or a wide area network. Or, the output of the doctor'scomputer can be stored on a portable memory device such as a universalserial bus (“USB”) drive that is taken and inserted into the renalfailure therapy machine. The first component is an estimation component.The second component is a prediction component. The third component isan optimization component. The output of the estimation component isused as an input to both the prediction and optimization components. Theprediction and optimization components can both be used to determinetherapy prescriptions that will yield suitable solute clearances, e.g.,for urea, beta 2-microglobulin (“β2-M”), phosphorus or phosphate (thetwo terms will be used interchangeably) and potassium. The doctor thenconsults with the patient to pick one or more chosen prescription thatthe patient believes best fits the patient's lifestyle.

As shown below, the inputs to the optimization component are the therapyoutcomes desired by the doctors. The output of the optimizationcomponent is one or more suitable therapy prescription for the patientto be run on a renal blood treatment machine, such as a hemodialysis,hemofiltration, or hemodiafiltration machine. The therapy prescriptioncan set therapy parameters, such as (i) therapy frequency, e.g., numberof therapies per week, (ii) therapy duration, e.g., one hour to eighthours, (iii) therapy type, e.g., single needle versus dual needle, (iv)dialysate flowrate and/or overall volume of fresh dialysate used duringtherapy, (v) blood flowrate, (vi) ultrafiltration rate and/or overallvolume of ultrafiltrate removed during therapy, (vii) dialysatecomposition, e.g., conductivity, and (viii) dialyzer or hemofilterparameters, such as dialyzer clearance capability or flux level.

The initial or estimation component includes a test that is run on thepatient while the patient is undergoing the therapy of choice, e.g.,hemodialysis, hemofiltration, or hemodiafiltration. The test uses a setof therapy prescription parameters, such as, treatment time, bloodflowrate and dialysate flowrate. While the patient is undergoingtreatment, blood samples are taken at various times over the treatment,e.g., at half-hour, forty-five minute or hour intervals. The samples areanalyzed to determine the level of certain therapy markers, such as ureaconcentration (small molecule), beta 2-microglobulin (“β2-M”) (middlemolecule), and phosphate or potassium (certain dialysis therapies canremove too much phosphate or potassium, so it is desirable to know ifthe patient may be predisposed to this phenomenon). In general, theconcentration of each of the markers will lower over time as the urea,β2-M, phosphate and potassium are cleared from the patient's blood.

The concentration or clearance results are then fed into a series ofmodels or algorithms for the estimation component to determine a set ofestimated patient parameters for (i) the particular patient, (ii) theparticular molecule and (iii) its corresponding algorithm. For example,one of the parameters is G, which is the generation rate for theparticular solute or molecule produced as a result of dietary intake,e.g., food and drink. K_(D), another estimated patient parameter, isdialyzer clearance for the molecule. K_(IC), a further estimatedparameter, is the patient's inter-compartmental mass transfer-areacoefficient for the molecule or solute. K_(M), another estimatedparameter, is the mobilization clearance of phosphorus or potassiumdetermining the rate at which phosphorus or potassium is released intothe extracellular space. V, yet another parameter that may be estimated,is the distribution volume of phosphorus or potassium. Other estimatedparameters are discussed below.

The prediction component uses the estimated patient parameters fed intothe modules or algorithms from the estimation component or module tocalculate clearance results for one or more solute, e.g., urea, β2-M,phosphate and/or potassium over a varied set of therapy prescriptionparameters. The prediction component also uses assumed patientparameters. For example, K_(NR), is the non-renal clearance of thesolute, which can be considered a constant, such that it does not haveto be estimated on an individual basis. Shown in detail below are graphsillustrating the output of the prediction component, in which acombination of therapy duration and therapy frequency is graphed alongthe x-axis and a solute concentration, e.g., urea, β2-M, phosphate orpotassium is graphed along the y-axis. The graphs (i) provide a visualcue to the average concentration level for the solute and (ii) estimatethe maximum concentration level that the solute will reach. The graphscan be used to determine one or more clinically acceptable parameters,such as standard Kt/V of urea and mean pre-treatment plasmaconcentration (“MPC”) of β2-M and pre-dialysis plasma concentration ofphosphorus and/or potassium, all of which in turn help identify theappropriate and customized therapy prescription. The doctor cancommunicate the acceptable prescriptions to the patient, who then picksone or more chosen prescription for downloading to the HHD machine.

The optimization component operates the reverse of the predicationcomponent and instead inputs desired solute concentration levels intothe models of the estimation component, using the estimated parametersof the estimation component to determine an optimized set of therapyprescription parameters for the patient. The optimization takes intoaccount the doctor's desired solute concentration clearance for thepatient for one or more solutes and the patient's preference, e.g., asto therapy frequency and duration. In one example, the optimizationcomponent inputs a doctor's requirement for urea, β2-M and phosphateclearances, which yields a plurality of therapy prescriptions that meetthe clearance requirements. The doctor and patient can then view theacceptable therapy prescriptions and select one or more chosenprescription for loading into the renal failure blood therapy, e.g.,HHD, machine.

Thus, the prediction and optimization components can both lead to chosentherapy prescriptions that are downloaded to the HHD machine. Theoptimization component may be easier to use to choose a suitable therapyprescription than the prediction components because it is less iterative(for the doctor) than the prediction component. However, the predictioncomponent can provide more detailed information for the doctor and for aparticular therapy prescription. Thus, in one particularly usefulimplementation of the present disclosure, the system estimates,optimizes and allows for choice and then predicts detailed results forthe optimized and chosen therapy prescription.

It is contemplated that the patient's estimated patient parameters,e.g., G, V_(D), K_(IC) and K_(M) be updated periodically to adjust for achanging condition of the patient and to adjust for actual data obtainedfrom past therapies. For example, the patient can have blood work doneperiodically, such that the downloaded prescription can be changed ifthe results of the blood work warrant such a change. Thus, the threecomponents can be cycled or updated periodically, e.g., once or twice ayear or as often as the doctor finds necessary.

In the second primary embodiment under Roman Numeral II below, thepresent disclosure sets forth methods of predicting serum phosphorusconcentrations in a patient during hemodialysis. In one embodiment, themethod includes measuring serum phosphorus concentrations (“C”) of thepatient over a hemodialysis treatment session time and anultrafiltration rate (“Q_(UF)”) calculated by a difference between pre-and post-dialytic body weight of the patient during an initialhemodialysis treatment session divided by a total treatment time of thetreatment session, estimating a phosphorous mobilization clearance(“K_(M)”) and a pre-dialysis distribution volume of phosphorus(“V_(PRE)”) for the patient, and predicting C of the patient at any timeduring any hemodialysis treatment session with the estimated K_(M) andV_(PRE) of the patient. C of the patient can be measured every 15 or 30minutes during the hemodialysis treatment session.

In a third primary embodiment under Roman Numeral III below, the presentdisclosure sets forth methods of predicting serum potassiumconcentrations in a patient during hemodialysis. In one embodimentsimilar to the method of predicting phosphorus concentrations in apatient during hemodialysis, the method includes measuring serumpotassium concentrations (also denoted as “C”) of the patient over ahemodialysis treatment session time and an ultrafiltration rate(“Q_(UF)”) calculated by a difference between pre- and post-dialyticbody weight of the patient during an initial hemodialysis treatmentsession divided by a total treatment time of the treatment session,estimating a potassium mobilization clearance (also denoted as “K_(M)”)and a pre-dialysis distribution volume of potassium (also denoted as“V_(PRE)”) for the patient, and predicting C of the patient at any timeduring any hemodialysis treatment session with the estimated K_(M) andV_(PRE) of the patient.

In another embodiment, a computing device for predicting serumphosphorus or potassium concentrations in a patient during hemodialysisincludes a display device, an input device, a processor, and a memorydevice that stores a plurality of instructions, which when executed bythe processor, cause the processor to operate with the display deviceand the input device to: (a) receive data relating to C of ahemodialysis patient over a hemodialysis treatment session time and anQ_(UF) calculated by a difference between pre- and post-dialytic bodyweight of the hemodialysis patient during a hemodialysis treatmentsession divided by a total treatment time of the treatment session, (b)estimate K_(M) and V_(PRE) for the patient, and (c) predict C of thepatient at any time during hemodialysis. The processor can operate withthe display device and the input device to receive data relating to atleast one of K_(R), K_(D) or a sampling time for collecting the serumphosphorus or potassium concentration.

In yet another embodiment, a method of determining steady state,pre-dialysis serum phosphorus or potassium levels in a hemodialysispatient includes obtaining a net generation of phosphorus or potassium(“G”) of the hemodialysis patient, determining steady state,pre-dialysis serum phosphorus or potassium levels (“C_(SS-PRE)”) of thehemodialysis patient, and simulating the effect of at least one of apatient parameter or a treatment parameter on C_(SS-PRE) of thehemodialysis patient.

In an embodiment, a computing device for determining steady state,pre-dialysis serum phosphorus or potassium levels in a hemodialysispatient includes a display device, an input device, a processor, and amemory device that stores a plurality of instructions, which whenexecuted by the processor, cause the processor to operate with thedisplay device and the input device to: (a) receive data relating to Gfrom at least a dietary phosphorus or potassium intake of a hemodialysispatient or a urea kinetic modeling of the hemodialysis patient, (b)determine C_(SS-PRE) of the patient, and (c) simulate the effect of atleast one of a patient parameter or a treatment parameter on theC_(SS-PRE) of the hemodialysis patient. The processor can operate withthe display device and the input device to receive data relating to atleast one of K_(R), K_(D), K_(M), V_(PRE), t_(tx), F, C_(PRE) about amonth before a hemodialysis treatment session or a sampling time forcollecting the serum phosphorus or potassium concentration. Thecomputing device can display a treatment regimen of the hemodialysispatient so that C_(SS-PRE) is within a desired range.

An advantage of the present disclosure is accordingly to provideimproved renal failure blood therapy systems and methods.

Another advantage of the present disclosure is to provide renal failureblood therapy systems and methods having a therapy prediction tool withwhich clinicians may adjust renal failure blood therapies, e.g., HHDtherapies, for specific patients with respect to the key soluteclearance measures.

Yet another advantage of the present disclosure is to provide renalfailure blood therapy systems and methods that offer clinicians multiplechoices to achieve desired target goals.

Still another advantage of the present disclosure is to provide renalfailure blood therapy systems and methods that employ a clinicallyviable and practical test procedure to help characterize the patient'sresponse to a particular renal blood therapy.

A further advantage of the present disclosure is to provide renalfailure blood therapy systems and methods that help to reduce the amountof trial and error in optimizing a blood therapy for the patient.

Still a further advantage of the present disclosure is to provide renalfailure blood therapy systems and methods that aid the patient inattempting to optimize lifestyle choices for therapy.

Another advantage of the present disclosure is to accurately predictplasma phosphorus or potassium levels in a patient during short,conventional, daily and nocturnal hemodialysis.

Yet another advantage of the present disclosure is to accurately predictsteady state, pre-dialysis plasma phosphorus or potassium serum levelsin a patient who is maintained by a hemodialysis therapy for a specifiedtime.

Still another advantage of the present disclosure is to develop ormodify hemodialysis treatment regimens so that the plasma phosphorus orpotassium serum levels of a hemodialysis patient are maintained within adesired range.

A further advantage of the present disclosure is to provide systems thatdevelop or modify hemodialysis treatment regimens so that the plasmaphosphorus or potassium serum levels of a hemodialysis patient aremaintained within a desired range.

Yet a further advantage of the present disclosure is to conservedialysate and other dialysis supplies, streamlining the therapy andlowering therapy cost.

Additional features and advantages are described herein, and will beapparent from, the following Detailed Description and the figures.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 is a schematic overview of one embodiment of a renal failureblood therapy system and method of the present disclosure.

FIG. 2 is an example of a system component selection screen of oneembodiment of a renal failure blood therapy system and method of thepresent disclosure.

FIG. 3 is an example of a patient parameter estimation input screen ofone embodiment of a renal failure blood therapy system and method of thepresent disclosure.

FIG. 4 is an example of a patient parameter estimation output screen ofone embodiment of a renal failure blood therapy system and method of thepresent disclosure.

FIG. 5 is an example of a therapy prediction input screen of oneembodiment of a renal failure blood therapy system and method of thepresent disclosure.

FIG. 6 is an example of a therapy prediction output screen of oneembodiment of a renal failure blood therapy system and method of thepresent disclosure.

FIG. 7 is an example of a therapy optimization input screen of oneembodiment of a renal failure blood therapy system and method of thepresent disclosure.

FIG. 8A is an example of a therapy optimization routine screen of oneembodiment of a renal failure blood therapy system and method of thepresent disclosure.

FIG. 8B is another example of a therapy optimization routine screen ofone embodiment of a renal failure blood therapy system and method of thepresent disclosure.

FIGS. 9A and 9B are a schematic flow diagram representing thecombination of FIGS. 9A and 9B and summarizing some of the features forthe renal failure blood therapy system and method of the presentdisclosure.

FIG. 10 illustrates an embodiment of a computing device of the presentdisclosure.

FIG. 11 is a conceptual description of the pseudo one-compartment model.

FIG. 12 shows the modeled and measured plasma phosphorus concentrationsfor patient 1 during short HD and conventional HD treatments.

FIG. 13 shows the modeled and measured plasma phosphorus concentrationsfor patient 2 during short HD and conventional HD treatments.

FIG. 14 shows the modeled and measured plasma phosphorus concentrationsfor patient 3 during short HD and conventional HD treatments.

FIG. 15 shows the modeled and measured plasma phosphorus concentrationsfor patient 4 during short HD and conventional HD treatments.

FIG. 16 shows the modeled and measured plasma phosphorus concentrationsfor patient 5 during short HD and conventional HD treatments.

FIG. 17 is a conceptual model used to describe steady state phosphorusmass balance over a time-averaged period.

FIG. 18 illustrates the effect of treatment frequency per se onpre-dialysis serum phosphorus concentration as a function of dialyzerphosphate clearance.

FIG. 19 shows the effects of an increase in treatment time and frequencywith reference to nocturnal forms of hemodialysis.

FIG. 20 shows the effect of increasing treatment frequency and treatmenttime relevant to short daily hemodialysis on pre-dialysis serumphosphorus concentration—K_(M)=50 ml/min.

FIG. 21 shows the effect of increasing treatment frequency and treatmenttime relevant to short daily hemodialysis on pre-dialysis serumphosphorus concentration—K_(M)=50 ml/min.

FIG. 22 shows the effect of increasing treatment frequency and treatmenttime relevant to short daily hemodialysis on pre-dialysis serumphosphorus concentration—K_(M)=150 ml/min.

FIG. 23 shows the effect of increasing treatment frequency and treatmenttime relevant to short daily hemodialysis on pre-dialysis serumphosphorus concentration—K_(M)=150 ml/min.

FIG. 24 shows the relationships between urea stdKt/v, β2-M MPC andtherapy duration.

FIG. 25 is a conceptual description of a pseudo one-compartment kineticmodel for potassium.

FIG. 26 shows predialysis potassium central distribution volume versuspotassium mobilization clearance based on experimental results.

FIG. 27 shows the dependence of serum potassium concentration versustime for four dialysate potassium concentration categories based onexperimental results.

FIG. 28 shows potassium mobilization clearance as a function of initialdialysate potassium concentration based on experimental results.

FIG. 29 shows predialysis potassium distribution volume as a function ofinitial dialysate potassium concentration based on experimental results.

FIG. 30 shows potassium removal divided by the predialysis serumconcentration for four dialysate potassium concentration categoriesbased on experimental results.

FIG. 31 shows normalized potassium removal versus serum potassiumconcentration based on experimental results.

DETAILED DESCRIPTION I. Therapy Estimation, Prediction and Optimization

Referring now to the drawings and in particular to FIG. 1, system 10illustrates one optimization system and method of the present disclosurefor implementing a therapy prescription into a renal failure bloodtherapy machine 100, such as a hemodialysis machine, and in particularhome hemodialysis (“HHD”) machine. One particularly well suited HHDmachine to operate the systems and methods of the present disclosure isset forth in the following United States patent applications: (i) U.S.Pub. No. 2008/0202591, (ii) U.S. Pub. No. 2008/0208103, (iii) U.S. Pub.No. 2008/0216898, (iv) U.S. Pub. No. 2009/0004033, (v) U.S. Pub. No.2009/0101549, (vi) U.S. Pub. No. 2009/0105629, U.S. Pub. No.2009/0107335, and (vii) U.S. Pub. No. 2009/0114582, the contents of eachof which are incorporated herein expressly by reference and relied upon.The HHD machine includes at least one processor and at least one memorythat are specifically modified or programmed to accept a therapyprescription to run, which is prescribed by a doctor, clinician or nurse(for convenience referred to collectively hereafter as a doctor 14unless otherwise specified). The therapy is downloaded to the HHDmachine (for convenience “HHD machine” refers collectively hereafter toa home hemodialysis, home hemofiltration, home hemodiafiltration or acontinuous renal replacement therapy (“CRRT”) machine unless otherwisespecified), e.g., via a memory storage device, such as a flash drive oruniversal serial bus (“USB”) drive, or via an internet or other local orwide area data network.

A patient 12 may be provided with multiple suitable therapies and beallowed to choose between the therapies, e.g., based upon the patient'sschedule that day or week. United States patent applications thatdisclose the provision of and selection from multiple peritonealdialysis therapies include: (i) U.S. Pub. No. 2010/0010424, (ii) U.S.Pub. No. 2010/0010423, (iii) U.S. Pub. No. 2010/0010426, (iv) U.S. Pub.No. 20100010427, and (v) U.S. Pub. No. 20100010428, each assigned to theassignee of the present disclosure, and the entire contents of each ofwhich are incorporated herein by reference and relied upon.

System 10 includes three components, namely, an estimation component 20,a prediction component 40 and an optimization component 60.

Estimation Component

Estimation component 20 includes a patient test 22, which involvesactually performing the therapy of choice, e.g., a hemodialysis,hemofiltration, hemodiafiltration or CRRT therapy on patient 12 andtaking blood samples at different times, e.g., at five different times,during the actual therapy. The test therapy can for example be afour-hour therapy, in which the sample times are, for example, at thebeginning of the test, ½ hour, one hour and two hours; or the testtherapy can for example be an eight-hour therapy, in which the samplesare taken at the beginning of therapy, and at one hour, two hours, fourhours, five hours, six hours and eight hours. The test therapy durationsand number of samples can be varied as desired.

The blood samples are each analyzed to determine the level ofconcentration of certain marker solutes, such as, urea 24 (smallmolecule), beta 2-microglobulin (“β2-M”) 26 (middle molecule), andphosphate 28. It is known that for certain patients running certaindialysis therapies, e.g., longer therapies, that too much phosphate canbe removed from the patient, to the point that in some instancesphosphate has to be returned to the patient. System 10 contemplates thedetermination of whether the patient may be predisposed to experiencinglow phosphate levels.

The concentration levels determined at known times during therapy arefed into a series of models or algorithms 30, one model for each soluteof concern, e.g., a first model 30 a for urea 24, a second model 30 bfor β2-M 26 and a third model 30 c for phosphate 28. A model forpotassium (not shown) can also be used. Models 30 (referringcollectively to models 30 a, 30 b, 30 c, . . . 30 n) can each includeone or more algorithm. Suitable models 30 are discussed herein.

The output of estimation component 20 includes a multitude of estimatedpatient parameters 32 for each solute 24, 26, and 28 of concern, whichare based on the patient's blood results using models 30 and aretherefore specific to the patient. While it is important to the presentdisclosure that the estimated patient parameters 32 are tailored to thepatient's physiologic make-up, the doctor may feel that a blood test istoo strenuous or invasive. Thus, system 10 also contemplates theestimating parameters 32 instead using empirical data, e.g., typicalparameter values for the patient based on for example age, weight, sex,blood type, typical blood pressure, height, therapy duration,nutritional status, and disease information related to the kinetics ofthe solute(s) of interest. It is believed that this data can bedeveloped over time using system 10. Important parameters for models 30of system 10, include estimated patient parameters 32 and known, assumedor calculated (outside of the model) values 42, such as:

K_(IC), which is the patient's inter-compartmental diffusion coefficientfor the molecule or solute and is an estimated parameter 32;

K_(D), which is a known dialyzer clearance for the particular moleculeor solute and can be a parameter 42 calculated outside the model;

K_(M), which is the patient's phosphorus or potassium mobilizationclearance and is an estimated parameter 32;

K_(NR), which is the patient's residual kidney coefficient for theparticular molecule or solute and may be a parameter 42 assumed to be aconstant;

V is the distribution volume of phosphorus or potassium and is anestimated parameter 32;

G, which is the generation rate for the particular solute or moleculeproduced by patient's intake and is an estimated parameter 32 or may bean assumed parameter 42;

V_(P), which is the perfused or extracellular volume, is an estimatedparameter 32;

V_(NP), which is the non-perfused or intracellular volume, is anestimated parameter 32;

V_(D), which is the solute distribution volume in the body, equal toV_(P)+V_(NP) for urea and beta2-microglobulin, is an estimated parameter32;

C_(P), which is the extracellular concentration of the solute, is aparameter 42 determined from test 22 and is therefore a known in themodels 30 of estimation component 20 (it should be noted that not onlycan C_(P) be a measured estimation component 20, C_(P) can also bepredicted along with C_(NP) by a prediction module, after which one cancompare measured and predicted versions of C_(P), etc., to gauge theperformance of system 10);

C_(NP), which is the intracellular concentration of the solute andcannot be measured, is not a result of test 22 and is not an input tothe models, it is instead a predicted output from the prediction andoptimization components;

Φ_(NP), which is a ratio of intracellular compartment volume to thetotal distribution volume and Φ_(P), which is the ratio of extracellularcompartment volume to the total distribution volume, both of which areknown parameters 42 from literature; and

α, which is an interdialytic fluid intake, i.e., water intake, is aparameter 42 calculated outside of models 30 based on average fluidintake, or weight gain.

As shown above, there are at least seven estimated parameters 32,namely, K_(IC), K_(M), V, G, V_(P), V_(NP), and V_(D), where V_(NP) andV_(D) are related to V_(D) through Φ_(P) and Φ_(NP). For convenience,only three of them are illustrated in FIG. 1, namely, K_(IC), V_(P) andV_(NP). It should be appreciated that any one, some or all of the sevenparameters can be estimated via estimation component 20 and models 30.It is contemplated to allow desired parameters for estimation to bechosen by the doctor, e.g., via selection boxes like those shown belowin FIG. 5 for the solutes under the prediction component 40.

There are also operational parameters 44 discussed herein, such as,blood flowrate, dialysate flowrate, dialysate total volume,ultrafiltration flowrate and ultrafiltration volume, and alsooperational parameters 44 affecting the patient's life style, such as:

T, which is in one instance the time duration at which each sample istaken in estimation component 20, and is therefore known parameter 42for the test 22 of estimation component 20, and is in another instancethe duration of dialysis in the prediction 40 and optimization 60components; and

F, which is the frequency of therapy, is taken to be one for the singletherapy of test 22, but is varied in the prediction 40 and optimization60 components.

Prediction Component

The estimated patient parameters 32 are then fed back into the models inprediction component 40, particularly into the personalized solute fluxand volume flux routine 50. Personalized solute flux and volume fluxroutine 50 uses in essence the same models or algorithms 30 of component20, but here using the estimated patient parameters 32 of estimationcomponent 30, as inputs, making the parameters 32 knowns instead ofvariables.

As shown in FIG. 1, the patient prediction component 40 accepts otherknown, assumed or calculated (outside of the model) values 42 for theparticular solute, such as dialyzer clearance K_(D), into personalizedsolute flux and volume flux routine 50. What is left as unknown withroutine 50 are (i) variable prescription operational parameters 44, suchas dialysis duration (“T”) and dialysis frequency (“F”) and (iii) soluteconcentrations 52 (C for both intracellular and extracellular) forsolutes 24, 26 and 28. Other machine operating parameters 44 that may beinputted into prediction component 40 and varied include blood flowrate,dialysate flowrate, dialysate total volume, ultrafiltration flowrate andultrafiltration volume. The solute distribution volume and total bodywater volume are not constant throughout therapy. System 10 accordinglyuses a variable-volume model allowing system 10 to change during thesimulated therapy duration. Prediction component 40 then computes C_(P),C_(NP), V_(P), and V_(NP) based on the given input variables such F, T,K_(D), etc. Patient prediction component 40 plugs in different realisticvalues for operational parameters 44, e.g., as x-axis values of a graph,and outputs solute concentrations 52, e.g., as y-axis values of thegraph.

The graphs allow the doctor to view how the concentration 52 of acertain solute varies over the course of, e.g., a week, for a particularset of prescription operational parameters 44. The doctor can see anaverage value or other accepted measure of quantification for theconcentration 52, e.g., a standardized Kt/v for the clearance of urea.The doctor can also see the peak value for the concentration 52 over thecourse of a therapy cycle.

The doctor can vary therapy duration and frequency input parameters todevelop multiple sets of graphs of the outputted solute concentrations,for example, set 1 of graphs for urea, β2-M and phosphate for therapyduration 1 and therapy frequency 1, and set 2 of graphs for urea, β2-Mand phosphate for therapy duration 2 and therapy frequency 2. Ifdesired, each set of graphs can be merged onto a single graph, e.g.,urea, β2-M and phosphate concentration on one single graph for therapyduration 1 and therapy frequency 1. The therapy duration(s) andfrequency(ies) that yield suitable solute concentrations can then becommunicated to the patient, who in turn applies life style preferences54 to yield one or more chosen therapy prescription 56 for downloadingto an HHD machine 100. The patient or doctor then selects one of theprescriptions, e.g., on a weekly basis to run for treatment. In otherexamples, blood and dialysate flowrates may also be adjusted to reachcertain clearance goals or to suit the patient's needs.

It is also expressly contemplated to optimize the visual outlay andfunctionality for the doctor, that is, to optimize the look andoperation of the graphs and tables, for example, to only allow valuesfor desired adequacy parameters to be manipulated. System 10 canmanipulate these values, so as to be customized to each doctor's needsand preferences. The screens shown herein are accordingly intended to beexamples. The examples are not intended to limit the invention.

Optimization Component

Optimization component 60 inputs a plurality of therapy targets 62, suchas target removal of urea 24, target removal of β2-M, target removal ofphosphate 28, and target removal of ultrafiltration (“UF”) or excesswater that has built inside the patient between treatments. Therapytargets 62 are entered into an optimization routine 70. In oneembodiment, optimization routine 70 uses in essence the kinetic modelsor equations 30 discussed above for estimation component 20, which likewith routine 50, have entered the estimated patient parameters 32obtained from estimation component 20. Then, calculations for eachsolute are made in the reverse of prediction component 40. That is,instead of entering prescription operational parameters 44 andcalculating solute concentration 52, a desired solute concentration 52is entered and operational parameters 44, which will satisfy the desiredor optimized solute concentration 52 are calculated. Here, the results72 of optimization component 60 are independent of, or more preciselythe reverse of, the results of prediction component 40. Optimizationroutine 70 identifies one or more therapy prescription 72 that meets thedesired or optimized solute concentration 52 for each designated solute.

In particular, the computational techniques using optimization routine70 of optimization component 60 have been found to be robust and stableprocedures that identify the therapy conditions that achieve the aimedinput target values (e.g. β2-M pre-dialysis serum concentration (“MPC”),urea standard Kt/v (std Kt/v), and phosphate steady state pre-dialysisserum concentration) by the clinician. The computational techniquesidentify multiple optimized therapy prescriptions and attempt to do soby performing the least number of iterative simulations. Outputtedtherapy parameters from optimization component 60 can include therapyduration (“T”), therapy frequency (“F”), blood and dialysate flow rates(“Q_(B)” and “Q_(D)”, respectively).

In one example, the target (i.e., input) urea stdKt/v and β2-M MPC maybe set to be 2.0 and 27.5 mg/L respectively. FIG. 24 shows therelationships (e.g., curves) between urea stdKt/v, β2-M MPC and therapyduration. In the example, the input target values are indicated bydotted lines. Optimization routine 70, following a relatively easy anditerative procedure, varies therapy duration T (for a given set of F,Q_(B), and Q_(D)) until both urea stdKt/v and β2-M MPC target values aresatisfied at the minimum necessary T, which is presumed to be theoptimal T for both the patient and the hemodialysis machine because thetime the patient is connected to the machine and the time the machineneeds to run and consume dialysate components is minimized.

In a first sample iteration, the optimization routine 70 performs asimulation at T₁=600 minutes, a therapy duration generally long enoughto produce adequacy parameters much better than desired. In a secondsample iteration, optimization routine 70 performs a simulation atT₂=600/2=300 minutes, producing satisfactory results. In the third step,optimization routine 70 performs a simulation at T₃=300/2=150 minutes,this time producing unsatisfactory results for both stdKt/v and β2-MMPC. In a fourth iteration, optimization routine 70 performs asimulation at an increased time T₄=(150+300)/2=225 minutes, producingsatisfactory result for stdKt/v only. In a fifth iteration, optimizationroutine 70 performs a simulation at a further increased timeT₅=(225+300)/2=263 minutes, producing satisfactory results for bothstdKt/v and β2-M MPC.

At the end of each step, if both target parameters are achieved, theoptimization routine 70 in one embodiment calculates the differencebetween the target and achieved values. If the difference for at leastone of the target parameters is greater than a threshold value, thenoptimization routine 70 performs yet another iteration to achieveresults closer to the target values, further lessening and optimizingduration T. Using the above procedure, optimization routine 70 performsa final simulation at T=(263+225)/2=244 minutes (bold vertical line),where both stdKt/v and β2-M MPC targets are satisfied and thedifferences between achieved stdKt/v and β2-M MPC and target stdKt/v andβ2-M MPC are small.

As illustrated, the optimal therapy duration T of 244 minutes is foundin only six iterations, again for a given set of F, Q_(B) and Q_(D).Multiple optimized therapy prescriptions can then be identified, e.g.,varying therapy duration, frequency, blood and/or dialysate flowrates,to allow the patient a choice based on lifestyle as discussed below.

Patient 12 and doctor 14 review the therapy prescriptions that meet thedesired or optimized solute concentration 52 and factor in the patient'slife style preferences 74. Perhaps patient 12 prefers a short dailytherapy during the day when the patient's spouse is awake forassistance. Or perhaps the patient works out on Monday, Wednesday andFriday and has less UF due to sweat on those days, preferring then torun treatments on other days.

Applying lifestyle preferences 74 to the therapy prescriptions 72 thatmeet the desired or optimized solute concentration 52 yields a chosenone or more therapy prescription 76. Chosen therapy prescription 56 and76 can be downloaded to machine 100, e.g., via manual entry into machine100, a download from a memory storage device, e.g., flash drive oruniversal serial bus (“USB”) drive, or a download from a data networksuch as an internet.

It is contemplated to modify a chosen one or more therapy prescription56 or 76 from time to time, e.g., due to regular and periodic bloodtesting 78, which the patient has performed from time to time. Thepatient may lose residual renal function over time causing the chosenone or more therapy prescription 56 or 76 to need to be modified. Theblood work may in any case indicate that the chosen one or more therapyprescription 56 or 76 is not removing one or more solute effectivelyenough, prompting a change. The patient may lose weight or undergo alifestyle change, which allows for a less rigorous one or more therapyprescription 56 or 76 to be used instead. In any case, it iscontemplated that lifestyle preferences 74 will continue to play a rolein potentially modifying the selected one or more therapy prescription76.

Sample Screen Shots

FIGS. 2 to 8B are sample screenshots further illustrating system 10described in connection with FIG. 1. The screen shots of FIGS. 2 to 8Bcan be custom generated per the request of the doctor and can beimplemented on the processing and memory of one or more computer used bythe doctor, clinician or nurse, which can be in data networkedcommunication with the HHD machine 100, e.g., via an internet, local orwide area network. It is also contemplated, especially for in-centermachines, to implement system 10 and the screen shots of FIGS. 2 to 8Bat the one or more processing and memory of machine 100.

FIG. 2 illustrates a sample startup screen that allows the doctor tochoose whether to enter or work with the patient parameter estimationcomponent 20, the therapy prediction component 40 or the therapyoptimization component 60. System 10 was described above from start tofinish. It may be however that patient parameter estimation component 20has already been performed, such that the doctor can jump directly toeither the prediction component 40 or the therapy optimization component60. As discussed above, therapy optimization component 60 can operateindependent of prediction component 40. Thus, it may be that the doctoronly uses one of prediction component 40 and optimization component 60at a particular time or application of system 10.

Prediction component 40 and therapy optimization component 60 rely oninformation from patient parameter estimation component 20, however, itshould be noted that if the patient 12 does not wish to undergo test 22,or the doctor 14 does not want the patient 12 to undergo test 22, it maybe possible, although not preferred, to use standardized values based onthe patient's information, such as age, sex, blood type, residual renalfunction if known, etc. It is also expressly contemplated to maintain adatabase of estimated patient parameters 32 developed over time usingsystem 10, which may provide viable standardized estimated patientparameters 32 based on patient category.

FIG. 3 illustrates a sample data entry and test 22 result screen forparameter estimation component 20. The screen at the left acceptspatient information, such as name, age, gender and weight. It iscontemplated for system 10 to be able to search for a file under any ofthis inputted data. The doctor enters data for test 22 into the middleof the screen of FIG. 3, such as total treatment time, blood flowrate,dialysate flowrate (or total volume) and UF rate or volume (notillustrated). The patient then undergoes a test therapy that is runaccording to this inputted data. The screen of FIG. 3 at the right thenaccepts the results of the blood testing done for urea 24, β2-M 26 andphosphate 28 at various times over the course of the treatment, forminga time-based profile for each of the analyzed solutes. The sample timesshow in FIG. 3 include a starting time, and one hour, two hours, fourhours, five hours, six hours and eight hours from the starting time.Other intervals including more or less time entries can be usedalternatively.

FIG. 4 illustrates a sample estimated patient parameters 32 displayscreen. Estimated patient parameters 32 can include, for example,generation rate G, intracellular clearance K_(IC), phosphorus orpotassium mobilization clearance K_(M), and distribution volume V_(D),(V_(D)=V_(P)+V_(NP), where V_(P) is the perfused or extracellularvolume, and V_(NP) is the non-perfused or intracellular volume). Thevalues for G, K_(IC), and V_(D), are the outputs of models 30 ofestimation components 20, and are then used as inputted data incomponents 40 and 60 as estimated patient parameters 32.

FIG. 5 illustrates a sample input screen for prediction component 40. Inthe illustrated example, the doctor at the left of the screen chooses torun the prediction routine for urea 24 and β2-M (boxes checked), but notfor phosphate 28 (boxes not checked). The doctor also enters operationalinputs 44, namely, the doctor wishes to model a therapy that is run fivedays a week (i.e., F) for three hours (i.e., T) per session. Asdiscussed above, other machine operating parameters that may be entered(and varied) alternatively or additionally to F and T include bloodflowrate, dialysate flowrate, dialysate total volume, ultrafiltrationflowrate and ultrafiltration total volume. “Back” and “Run” buttonsallow the doctor to navigate through each of the components 20, 40 or 60when the particular component are selected.

FIG. 6 illustrates a sample output screen for prediction component 40,which shows the results after running the personalized solute flux andvolume equations 50. If desired, the concentration results can be mergedonto a single graph, for example, with the urea concentration scalealong the left and β2-M scale along the right. Solute concentrations 52could alternatively be displayed in spreadsheet form but are shown inFIG. 6 in graphical form, with days from a start of using the particulartherapy prescription along the x-axis. This way, the doctor can readilysee the predicted solute profile for a given frequency and duration, andfor the patient's personalized estimated parameters. Soluteconcentrations 52 are also shown in an average or standardized form,e.g., as a standard Kt/v, which is understood by those of skill in theart. Knowing peak concentration and the average or standardizedconcentration, the doctor can quickly determine if the proposedfrequency and duration are adequate for the selected solutes, here, urea24 and β2-M 26. As delineated in FIG. 6, P is for perfused orextracellular, NP is for non-perfused or intracellular. If the solute,for example urea, is in the extracellular or blood volume then thedialyzer can readily clear the solute. If the solute is in theintracellular volume, then the solute has to first pass in to theextracellular volume overcoming the resistance defined by K_(IC).

FIG. 6 illustrates concentration values for a particular therapyduration T and frequency F. It is contemplated for the doctor to re-runprediction component 40 to vary T and F. The doctor can then choose oneor more sets of graphs, e.g., from (i) T₁, F₁; (ii) T₂, F₂; (iii) T₃,F₃; etc., that are clinically acceptable. The acceptable graphs or theircorresponding therapy prescriptions can then be reviewed with thepatient, who selects one or more graph or prescription that best meetsthe patient's lifestyle needs and/or requirements.

FIG. 7 illustrates a sample input screen for optimization component 60.Running opposite to prediction component 40, the doctor in optimizationcomponent 60 enters desired values for therapy results, e.g., a desiredvalue for urea, e.g., via standard Kt/v, a desired value for phosphorusin, e.g., pre-dialysis phosphorus plasma concentration in milligrams perliter, a desired value for β2-M in, e.g., milligrams per liter, and adesired ultrafiltrate (“UF”) removal value, e.g., in liters. UF isgenerally a machine-controlled function but can affect solute removal,so the input of UF is desirable for optimization.

FIG. 8A shows an example of an optimization routine 70, which forinputted urea and β2-M (and phosphate if desired) requirements, shows aspreadsheet of frequency in Days Per Week (along the side) and TherapyDuration in hours (along the top), and places an “X” in the cellcorresponding to a treatment that will meet the requirement for thatsolute. In the twelve possible combinations shown in FIG. 8, two (fourdays of three hour treatments and vice versa) meet the requirements forurea and β2-M requirements. The patient can then decide which optionfits his/her lifestyle better. Or, both prescriptions can be machineentered or chosen prescriptions 72. The patient then decides, forexample on a weekly basis, which of the two approved and chosenprescriptions is a better fit for that week.

FIG. 8A also shows example inputted parameters 72 for optimizationcomponent 60, here, resulting blood flowrate, total solution volume,dialysate flowrate (or total volume) and UF rate (or volume), which areused in the equations for all of the frequency combinations in the ureaand β2-M optimization routines 70. The doctor, independent of system 10,may calculate blood flowrate and dialysate flowrate, etc., to achievedesired K_(D) values. However, those calculations are independent of thecalculations taking place as part of the prediction 40 and/oroptimization 60 components. For system 10, flowrates are inputs todetermine K_(D), which in turn is an input to the prediction and/oroptimization components 40 and 60.

FIG. 8B shows another example of an optimization routine 70, which forinputted urea and β2-M (and phosphate if desired) requirements, shows aspreadsheet of Therapy Frequency (i.e., F) in days of the week (alongside) and Therapy Duration (i.e., T) in hours (along top). Here, theactual days of the week are shown. Optimization component 60 can discernbetween different combinations of the same number of days, e.g., threedays Monday/Wednesday/Friday versus three daysMonday/Wednesday/Saturday. In one embodiment, system 10 assumes certainpreset days when therapy frequency values are entered. For instance, foran F of three days per week, system 10 would assume, e.g.,Monday/Wednesday/Friday. System 10, however, allows doctor 14 to enterspecific days (as opposed to entering F). System 10 makes thecalculations according to the days entered. The ability of simulatingcustom therapy days can be important because system 10 can then moreaccurately track the accumulation of solutes within the body.

In FIG. 8B, each cell is then color coded or otherwise designated intoone of three categories (for example) for the clearance of not just aparticular solute, but for the analyzed solutes as a group. The desiredstandardized Kt/v values for urea located at the upper right of thechart of routine 70 show the group of solute cleaners. Each cell ofsolute clearances is labeled in the example as inadequate, borderline oradequate. For example, adequate can mean meets all requirements,borderline can mean meets some requirements or almost meets allrequirements, while inadequate means misses most or all requirements.More or less than three classifications having different meanings can beused alternatively. The patient can then choose from one of the adequatetherapy prescription cells, for example, choose the least rigoroustherapy prescription.

Referring now to FIGS. 9A and 9B, method 110 illustrates therelationship between components 20, 40 and 60 of system 10 discussedherein. Method 110 is meant to help understand the interrelationshipbetween components 20, 40 and 60 and is in no way meant to describe allthe alternatives for the components, which have been described in detailabove.

At oval 112, method 110 begins. At block 114, the test therapy isperformed on the patient to determine concentration levels for varioussolutes, such as urea, β2-M and phosphate. The solutes for system 10 andmethod 110 are not limited to those three solutes and could includeothers, such as potassium, calcium, parathyroid hormone (“PTH”), andcertain protein-bound solutes such as p-cresol sulfate. It is expresslycontemplated to include these and other solutes in system 10 and method110. The additional solutes can at least be tracked via the testing inestimation component 20 for example as they relate to general adequacyand/or to correlate with phosphate clearance/mobilization. Theadditional solutes can eventually be predicted via prediction component40 and optimized via component 60 when models are devised in the futurefor the additional solutes, as has been done below for phosphate.

At blocks 116 a, 116 b, and 116 c, the concentration levels for urea,β2-M and phosphate are fed into the corresponding kinetic model todetermine at least one patient-specific parameter. All of the testconcentrations of the solute, less than all of the test concentrationsof the solute, or some averaged test concentration for the solute may beentered into the corresponding kinetic model.

At blocks 118 a, 118 b, and 188 c, the at least one patient-specificestimated parameter is fed along with at least one machine operationalparameter, such as therapy duration (i.e., T) and therapy frequency(i.e., F), into the corresponding kinetic model for urea, β2-M andphosphate to determine clearance volumes or solution levels for thesolute.

At blocks 120 a, 120 b, and 120 c, the solute clearance values for urea,β2-M and phosphate are graphed (could be single combined graph) ortabulated for the doctor's evaluation. At diamonds 122 a, 122 b and 122c, it is determined whether blocks 118 a to 118 c and 120 a to 120 c arerepeated for another set of inputted operational therapy parameters. Ifnot, at block 124 the doctor determines which graph(s), tabulation(s),prescription(s) are clinically acceptable.

At blocks 126 a to 126 c, the patient-specific estimated parameters forurea, β2-M and phosphate are fed into the corresponding kinetic modelalong with a desired solute removal level or level for the solute todetermine one or more machine operational parameter that will satisfythe equation and achieve the desired level for the one or more solute.

At block 128, the machine operating parameters that achieve the desiredsolute level (or some of the desired solute levels) are tabulated forthe doctor. The doctor can then hone in on the best clinicallyacceptable therapies for the patient.

At block 130, which is fed from both the predicting block 124 and theoptimization block 128, the doctor consults with the patient todetermine which one or more chosen therapy prescription best suits thepatient's personal needs. As discussed above, machine operationalparameters include T, F and others, such as fluid flowrates and/orvolumes. These other parameters are likely to be mandated by the doctorand not be as negotiable with the patient. To a certain degree, T and Fwill drive the other parameters. For example, a shorter therapy willlikely require higher flowrates.

At block 132, the chosen one or more therapy prescription is downloadedto the patient's (or clinic's) renal failure therapy machine (e.g., HHDmachine 100). If multiple chosen prescriptions are downloaded, thepatient may be empowered to choose, e.g., weekly, which prescription torun. Alternatively, the doctor may dictate, at least initially, whichprescription to run. The prescription download may be via a data networkor data storage device such as USB or flash drive.

It should be appreciated that system 10 in alternative embodiments canincorporate any of the methods, models and equations discussed below andin detail in sections II and III.

Kinetic Modeling (i) Urea and β2-M Modeling

Suitable kinetic models 30 for urea and β2-M for system 10 are shownbelow and described in detail in by: Ward (Ward et al., KidneyInternational, 69: 1431 to 1437 (2006)), the entire contents of whichare incorporated expressly herein by reference and relied upon, for atwo compartment model,

$\frac{\left( {V_{P}C_{p}} \right)}{t} = {{\Phi_{p}G} + {K_{ic}\left( {C_{np} - C_{p}} \right)} - {\Theta \; K_{d}C_{p}} - {K_{nr}C_{p}} + {{\Theta\Phi}_{np}Q_{uf}C_{np}} - {\left( {1 - \Theta} \right)\Phi_{np}\alpha \; C_{p}}}$$\frac{\left( {V_{np}C_{np}} \right)}{t} = {{\Phi_{np}G} + {K_{ic}\left( {C_{p} - C_{np}} \right)} - {{\Theta\Phi}_{np}Q_{{uf}\;}C_{np}} + {\left( {1 - \Theta} \right)\Phi_{np}\alpha \; C_{p}}}$

and Clark (Clark et al., J. AM. Soc. Nephrol., 10: 601-609 (1999)), theentire contents of which are incorporated expressly herein by referenceand relied upon,

$\frac{\left( {V_{p}C_{p}} \right)}{t} = {G + {K_{ic}\left( {C_{np} - C_{p}} \right)} - {\Theta \; K_{d}C_{p}} - {K_{nr}C_{p}}}$$\frac{\left( {V_{np}C_{np}} \right)}{t} = {K_{ic}\left( {C_{p} - C_{np}} \right)}$$\frac{\left( V_{p} \right)}{t} = {{{- \Theta}\; \Phi_{p}Q_{uf}} + {\left( {1 - \Theta} \right)\Phi_{p}\alpha}}$$\frac{\left( V_{np} \right)}{t} = {{{- {\Theta\Phi}_{np}}Q_{uf}} + {\left( {1 - \Theta} \right)\Phi_{np}\alpha}}$

The above equations are applicable to both urea and β2-M. The same modelis used for both solutes, with the parameter values being different,such as, generation rate, non-renal clearance, distribution volume, etc.

(ii) Mass Balance Modeling

One suitable model for electrolyte balance for system 10, e.g., forsodium, potassium, etc., is a three compartment model, and is describedin detail by Ursino et al., (Ursino M. et al., Prediction of SoluteKinetics, Acid-Base Status, and Blood Volume Changes During ProfiledHemodialysis, Annals of Biomedical Engineering, Vol. 28, pp. 204-216(2000)), the entire contents of which are incorporated expressly hereinby reference and relied upon.

(iii) Modeling Modifications for Replacement Fluid

As described herein, system 10 is not limited to dialysis and can beapplied to other renal failure blood treatments such as hemofiltrationand hemodiafiltration. Hemofiltration and hemodiafiltration both involvethe use of a replacement fluid, which is pumped directly into the bloodlines for convective clearance used in place of (hemofiltration) or inaddition to (hemodiafiltration) the osmotic clearance of dialysis.

The equations below show modifications to the kinetic models thatApplicants have found can be made to all for use of replacement fluid.The first equation below shows the effect on mass balance. Inparticular, the factor Q_(R)*C_(s,R) is added for replacement fluid. Thesecond equation below shows the effect of convective clearance(J_(s)(t)) on dialyzer clearance.

$\frac{{M_{s,{ex}}(t)}}{t} = {{- {\Phi_{s}(t)}} - {J(t)} + {R_{s,{ex}}(t)} + {Q_{R} \cdot c_{s,R}}}$${J_{s}(t)} = {S_{s} \cdot {Q_{F}(t)} \cdot \frac{c_{s,p}(t)}{r}}$

II. Phosphate Modeling Phosphate Prediction Methods in HemodialysisPatients and Applications Thereof

In light of the systems discussed herein, it is contemplated to providemethods of predicting serum or plasma phosphorus concentrations orlevels in a hemodialysis patient before, during and after hemodialysistherapies. Being able to predict serum phosphorus levels can be usefulin determining optimal treatment regimens for hemodialysis patients.These methods can be incorporated into any of the systems and computingdevices described herein to optimize hemodialysis therapies for thepatient.

Elevated levels of serum phosphorus in end stage renal disease patientshave been associated with greater risk of mortality, primarily due tocardiac-related causes. Such associations have been demonstrated amongvarious countries throughout the world and over time. Although thephysiological mechanisms involved remain incompletely understood,inadequate control of serum phosphorus levels and the use ofcalcium-based phosphate binders have been linked to rapid progression ofcoronary calcification, increased stiffness of the arterial wall andhigh blood pressure.

Control of serum phosphorus concentrations in most hemodialysis (“HD”)patients requires both the daily use of oral phosphate binders toinhibit intestinal absorption of phosphate and the removal of phosphateby HD treatments. Despite this dual approach, hyperphosphatemia oftenoccurs because typical western diets contain high phosphate content.Calcium-based oral phosphate binders are still used extensively becauseof their low cost, and more effective binders are under activedevelopment. Other efforts have attempted to increase dialytic removalof phosphorus during thrice-weekly therapy by various methods, oftenwithout substantial improvements. The only HD prescription parameterthat has been shown to consistently reduce serum phosphorusconcentrations is the use of longer treatments, both duringthrice-weekly HD and during HD treatments applied more frequently.

Methods of predicting or determining serum phosphorus levels of apatient undergoing hemodialysis using a robust and practical phosphoruskinetic model allow for effectively modifying new HD treatmentmodalities on an individual patient basis. In an embodiment, a method ofpredicting serum phosphorus concentration in a patient duringhemodialysis is provided. The method includes measuring serum phosphorusconcentrations (“C”) of the patient over a hemodialysis treatmentsession time (using any suitable methods of measuring such as, e.g.,fluorometric and colorimetric assays) and an ultrafiltration or fluidremoval rate (“Q_(UF)”) calculated by a difference between pre- andpost-dialytic body weight of the patient during an initial hemodialysistreatment session divided by the total treatment time of the treatmentsession and estimating K_(M) and V_(PRE) for the patient using anon-linear least squares fitting to the governing transport equationshaving analytical solutions of the following form:

$\begin{matrix}{{{C(t)} = {C_{PRE}\left\lbrack \frac{K_{M} + {\left( {K_{D} + K_{R} - Q_{UF}} \right)\left( \frac{V(t)}{V_{PRE}} \right)^{\frac{K_{M} + K_{D} + K_{R} - Q_{UF}}{Q_{UF}}}}}{K_{M} + K_{D} + K_{R} - Q_{UF}} \right\rbrack}}\mspace{79mu} {and}} & \left( {{II}\text{-}A} \right) \\{\mspace{79mu} {{C(T)} = {C_{PRE} - {\left( {C_{PRE} - C_{POST}} \right)^{({- \frac{K_{M}T}{V_{PRE} - {Q_{UF}t_{tx}}}})}}}}} & \left( {{II}\text{-}B} \right)\end{matrix}$

wherein t is a time during the hemodialysis treatment session, T is atime after an end of the hemodialysis treatment session, t_(tx) is atotal duration of the hemodialysis treatment session, C_(PRE) is apre-dialysis plasma phosphorus concentration, C_(POST) is apost-dialytic plasma phosphorus concentration, K_(M) is a phosphorousmobilization clearance of the patient, K_(R) is a residual renalclearance of phosphate, K_(D) is a dialyzer phosphate clearance, V_(PRE)is a pre-dialysis distribution volume of phosphorus of the patient, and

V(t)=V _(PRE) −Q _(UF) ×t  (II-C).

C (i.e., serum phosphorus concentrations) of the patient can then bepredicted at any time during any hemodialysis treatment session by usingthe equations II-A and II-B, for the previously estimated set of K_(M)and V_(PRE) of the patient. Alternative to the non-linear least squaresfitting, V_(PRE) can also be estimated as a certain percentage of bodyweight or body water volume of the patient.

In another embodiment, a method of predicting serum phosphorusconcentration in a patient during hemodialysis is provided when theultrafiltration rate is assumed to be negligible (i.e., Q_(UF)=0). Themethod includes measuring C of the patient during an initialhemodialysis treatment session (using any suitable methods of measuringsuch as, e.g., fluorometric and colorimetric assays) and estimatingK_(M) and V_(PRE) for the patient using a non-linear least squaresfitting to the governing transport equations having analytical solutionsof the following form:

$\begin{matrix}{{{C(t)} = {C_{PRE}\left\lbrack \frac{K_{M} + {\left( {K_{D} + K_{R}} \right)^{{- t}\frac{({K_{M} + K_{D} + K_{R}})}{V_{PRE}}}}}{K_{M} + K_{D} + K_{R}} \right\rbrack}}{and}} & \left( {{II}\text{-}D} \right) \\{{C(T)} = {C_{PRE} - {\left( {C_{PRE} - C_{POST}} \right)^{({- \frac{K_{M}T}{V_{PRE}}})}}}} & \left( {{II}\text{-}E} \right)\end{matrix}$

C of the patient can be predicted at any time during any hemodialysistreatment session by using the equations II-D and II-E for a given setof previously estimated parameters, K_(M) and V_(PRE), of the patient.Alternatively, V_(PRE) can be further estimated as a certain percentageof body weight or body water volume of the patient. In an embodiment,K_(M) may be estimated using data from a case where Q_(UF)≠0 and used inequation D where Q_(UF)=0.

In any of the methods of predicting serum phosphorus concentrations in apatient during hemodialysis described herein, K_(D) can be determinedusing the equation:

$\begin{matrix}{\mspace{79mu} {{K_{D} = {Q_{B}\frac{\left( {0.94 - {{Hct} \times 100}} \right)\left( {^{Z} - 1} \right)}{\left( {^{Z} - \frac{\left( {0.94 - {{Hct} \times 100}} \right)Q_{B}}{Q_{D}}} \right)}}}\mspace{79mu} {wherein}}} & \left( {{II}\text{-}F} \right) \\{\mspace{79mu} {{Z = {K_{o}A\; \frac{\left( {Q_{D} - {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B}}} \right)}{\left( {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B} \times Q_{D}} \right)}}},}} & \left( {{II}\text{-}G} \right) \\{{{K_{o}A} = {\frac{\left( {0.94 - {{Hct} \times 100}} \right)Q_{B,M} \times Q_{D,M}}{Q_{D,M} - {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B,M}}} \times {\ln\left( \frac{1 - {K_{D,M}/Q_{D,M}}}{1 - {K_{D,M}/\left\lbrack {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B,M}} \right\rbrack}} \right)}}},} & \left( {{II}\text{-}H} \right)\end{matrix}$

Q_(B) and Q_(D) are the blood and dialysate flow rates at which thedesired dialyzer clearance K_(D) is calculated using equations II-F andII-G. K_(O)A is a dialyzer mass transfer area coefficient for phosphateobtained as a result of a previous measurement where the set of bloodand dialysate flow rates Q_(B,M) and Q_(D,M) resulted in dialyzerclearance K_(D,M), and Hct is hematocrit count measured from patient'sblood sample. Alternatively, K_(D) can be determined at any time t usingthe equation:

$\begin{matrix}{K_{D} = \frac{{C_{D}\left( t_{s} \right)}{Q_{D}\left( t_{s} \right)}}{C\left( t_{s} \right)}} & \left( {{II}\text{-}I} \right)\end{matrix}$

wherein t_(s) is a sampling time and C_(D)(t_(s)) is a concentration ofphosphorus in a dialysate outflow at time t_(s), Q_(D)(t_(s)) is adialysate flowrate at time t_(s), and C(t_(s)) is a serum phosphorusconcentration at time t_(s).

Alternative to non-linear least squares fitting, K_(M) can be determinedusing the following algebraic equation:

$\begin{matrix}{K_{M} = {{C_{POST}\left( \frac{K_{D} - Q_{UF}}{C_{PRE} - C_{POST}} \right)}.}} & \left( {{II}\text{-}J} \right)\end{matrix}$

C of the patient can be measured at any suitable time during thehemodialysis treatment session, for example, such as every 15 or 30minutes. t_(tx) can be any suitable amount of time such as, for example,2, 4 or 8 hours. T can be any suitable time, for example, such as 30minutes or 1 hour.

V_(POST) is a measure of the distribution volume of phosphorus at theend of the hemodialysis treatment when the patient is considered to benormohydrated. This parameter approximates the volume of extracellularfluids. Thus, V_(POST) is a clinically relevant patient parameter thatcan be used to evaluate the patient's hydration status. In anapplication from knowing the previously determined V_(PRE), V_(POST) canbe determined using the equation:

V _(POST) =V _(PRE) −Q _(UF) ×t _(tx)  (II-K)

and a suitable therapy can be provided to the patient based on the valueof V_(POST). As seen from equation II-K, if Q_(UF)=0, thenV_(POST)=V_(PRE).

Specific steps of the methods of predicting phosphorus mobilization in apatient during hemodialysis can be performed using a computing device.Such a computing device can include a display device, an input device, aprocessor, and a memory device that stores a plurality of instructions,which when executed by the processor, cause the processor to operatewith the display device and the input device to (a) receive datarelating to C of a hemodialysis patient over a hemodialysis treatmentsession time and a Q_(UF) calculated based on a difference between pre-and post-dialytic body weight of the hemodialysis patient during ahemodialysis treatment session divided by the total treatment time ofthe treatment session; (b) estimating K_(M) and V_(PRE) for the patientusing a non-linear least squares fitting to the governing transportequations having analytical solutions of the following form:

$\begin{matrix}{{{C(t)} = {C_{PRE}\left\lbrack \frac{K_{M} + {\left( {K_{D} + K_{R} - Q_{UF}} \right)\left( \frac{V(t)}{V_{PRE}} \right)^{\frac{K_{M} + K_{D} + K_{R} - Q_{UF}}{Q_{UF}}}}}{K_{M} + K_{D} + K_{R} - Q_{UF}} \right\rbrack}}{and}} & \left( {{II}\text{-}L} \right) \\{{{C(T)} = {C_{PRE} - {\left( {C_{PRE} - C_{POST}} \right)^{({- \frac{K_{M}T}{V_{PRE} - {Q_{{UF}\mspace{11mu}}t_{tx}}}})}}}};} & \left( {{II}\text{-}M} \right)\end{matrix}$

and (c) predict C of the patient at any time during hemodialysis byusing the equations II-L and II-M for a given set of estimatedparameters, K_(M) and V_(PRE), of the patient. It should be appreciatedthat the variables for equations II-L and II-M can be determined usingany of the equations set forth herein. The information/data obtained forthe hemodialysis patient can be displayed/printed out and used by thehealthcare provider to provide improved treatment and nutritionalregimens for the hemodialysis patient. Any of the unknown factors can bedetermined using the appropriate equations or measurements discussedpreviously for the methods of determining phosphorus mobilization in apatient during hemodialysis.

If Q_(UF)=0, then

$\begin{matrix}{{C(t)} = {{C_{PRE}\left\lbrack \frac{K_{M} + {\left( {K_{D} + K_{R}} \right)^{{- t}\frac{({K_{M} + K_{D} + K_{R}})}{V_{PRE}}}}}{K_{M} + K_{D} + K_{R}} \right\rbrack}.}} & \left( {{II}\text{-}N} \right)\end{matrix}$

The computing device can also be preprogrammed or run according tosoftware that causes the processor to operate with the display deviceand the input device to receive data relating to at least one of K_(R),K_(D) or a sampling time for collecting the serum phosphorusconcentration. In an embodiment, the computing device can be system 10described in section I.

Along with the previously described methods of determining phosphorusmobilization in a patient during hemodialysis, a mass balance model topredict steady state, pre-dialysis serum phosphorus levels(“C_(SS-PRE)”) in patients treated with HD therapies has also beendeveloped. A mass balance model was used in combination with a pseudoone-compartment model for intradialytic and rebound periods to determinesteady state pre-dialysis serum phosphorus levels in individualpatients. Using this model, the effect of specific therapy parameters(e.g., dialyzer phosphate clearances, weekly therapy frequency, therapyduration, etc.) on individual hemodialysis patients' serum phosphoruslevels can be evaluated.

The disclosed steady state, mass balance model combines theintradialytic phosphorus kinetics with dietary intake, use of phosphatebinders, and residual renal clearance to predict steady state,pre-dialysis serum phosphorus levels. Unlike those with previous models,the predictions with this model involve simplified calculations; hence,this model can easily be integrated in daily clinical practice.Furthermore, the model involves patient-specific parameters enablingindividualized predictions. This model can eventually be used tooptimize therapies with a HHD device to remove adequate amounts ofphosphorus using minimum necessary volumes of dialysate (i.e., minimizedwater consumption). Alternatively, the model can be used to determinethe amount of required phosphate salt supplements in dialysate.

In an application of the kinetic model, a method of determining theC_(SS-PRE) in a hemodialysis patient is provided. The method includesobtaining a net generation of phosphorus (“G”) from at least a dietaryphosphorus intake of the patient or a urea kinetic modeling of thepatient and determining C_(SS-PRE) of the hemodialysis patient using theequation:

$\begin{matrix}{C_{{SS} - {PRE}} = \frac{(G)\left( {10080/F} \right)}{{\left( {K_{D} + K_{R}} \right)\mspace{11mu} n\; {\overset{\_}{C}}_{tx}t_{tx}} + {K_{R}n\; {{\overset{\_}{C}}_{i}\left( {{10080/F} - t_{tx}} \right)}}}} & \left( {{II}\text{-}O} \right)\end{matrix}$

wherein F is a frequency of treatments per week, t_(tx) is a treatmenttime for one hemodialysis treatment session (e.g., in units of minutesper treatment session), K_(D) is a dialyzer phosphate clearance, K_(R)is a residual renal clearance of phosphate, n C _(tx) is the normalizedtime averaged plasma phosphorus concentration during a dialysistreatment, and n C _(i) is the normalized time averaged plasmaphosphorus concentration for an interdialytic interval. The effect of atleast one of a patient parameter or a treatment parameter on C_(SS-PRE)of the patient can be simulated so as to obtain an optimal range ofC_(SS-PRE) for the patient. For example, the patient parameter can be G,K_(M) or V_(PRE), and the treatment parameter can be t_(tx), K_(D)(e.g., Q_(B), Q_(D)) or F.

In an alternative embodiment, a method of predicting the C_(SS-PRE) in ahemodialysis patient is provided. The method includes determining a netgeneration of phosphorus (“G”) using the equation:

$\begin{matrix}{G = {C_{{SS} - {PRE} - {IN}} \times \left\lbrack \frac{{\left( {K_{D} + K_{R}} \right)\; n\; {\overset{\_}{C}}_{tx}t_{tx}} + {K_{R}n\; {{\overset{\_}{C}}_{i}\left( {{10080/F} - t_{tx}} \right)}}}{10080/F} \right\rbrack}} & \left( {{II}\text{-}P} \right)\end{matrix}$

wherein C_(SS-PRE-IN) is an initial, measured, steady state,pre-dialysis serum phosphorus level of the hemodialysis patient who ismaintained by a hemodialysis therapy (e.g., identified by K_(D), F andt_(tx)) for a specified time prior to the calculation of G usingequation P. The specified time can be, for example, at least one week,two weeks, three weeks, 1 month, 2 months, 3 months, 4 months or moreprior to the time when G is calculated. F is a frequency of treatmentsper week, t_(tx) is a treatment time for one hemodialysis treatmentsession (e.g., in units of minutes per treatment session), K_(D) is adialyzer phosphate clearance, K_(R) is a residual renal clearance ofphosphate, n C _(tx) is the normalized time averaged plasma phosphorusconcentration during a dialysis treatment, and n C _(i) is thenormalized time averaged plasma phosphorus concentration for aninterdialytic interval.

Once G has been calculated using equation II-P or estimated by othermethods, it can be used in predicting the effect of changes inhemodialysis treatment parameters on the steady state serum phosphorusconcentration. For example, once G of the hemodialysis patient is known,C_(SS-PRE) of the patient under different hemodialysis treatmentconditions can be predicted by rearranging equation II-P to formequation II-O and utilizing the known G to solve for C_(SS-PRE) of thehemodialysis patient. The effect of at least one of a patient parameteror a treatment parameter on C_(SS-PRE) of the patient can be simulated,and a treatment regimen of the hemodialysis patient can then be modifiedso that C_(SS-PRE) is within a desired range.

In general, there is an optimal range of steady state, pre-dialysisserum phosphorus levels in patients with end stage renal disease.Optimal prescription/regimen/nutritional therapies resulting in steadystate, pre-dialysis phosphorus levels within the desired optimal rangescan be determined using equations II-O and II-P, for example, in theOptimization Component of the HHD system previously described herein.Because changes in the hemodialysis prescription or in patient behavior(e.g., changes in diet) could lead to changes in G, optimization of homehemodialysis therapies based on equations II-O and II-P to maintainC_(SS-PRE) within a desired range is advantageous.

In any of the methods of determining the G or the C_(SS-PRE) in ahemodialysis patient, n C _(tx) and n C _(i) can be determined using theequations:

$\begin{matrix}{{n\; {\overset{\_}{C}}_{tx}} = {\frac{1}{t_{tx}}\left\{ {{\left\lbrack \frac{K_{M}t_{tx}}{K_{M} + K_{R} + K_{D} - Q_{UF}} \right\rbrack + {\left\lbrack {1 - \frac{K_{M}}{K_{M} + K_{R} + K_{D} - Q_{UF}}} \right\rbrack \left. \quad{\left\lbrack \frac{V_{PRE}}{K_{M} + K_{R} + K_{D}} \right\rbrack \left\lbrack {1 - \left( \frac{V_{POST}}{V_{PRE}} \right)^{{({K_{M} + K_{R} + K_{D}})}/Q_{UF}}} \right\rbrack} \right\}}},\mspace{79mu} {and}} \right.}} & (Q) \\{{n\; {\overset{\_}{C}}_{i}} = {\frac{1}{{10080/F} - t_{tx}}\left\{ {{\left\lbrack \frac{K_{M}\left( {{10080/F} - t_{tx}} \right)}{K_{M} + K_{R} + Q_{WG}} \right\rbrack + {\left\lbrack {\frac{K_{M}}{K_{M} + K_{R} + K_{D} - Q_{UF}} + {\left\lbrack {1 - \frac{K_{M}}{K_{M} + K_{R} + K_{D} - Q_{UF}}} \right\rbrack \left\lbrack \frac{V_{POST}}{V_{PRE}} \right\rbrack}^{{({K_{M} + K_{D} + K_{R} - Q_{UF}})}/Q_{UF}} - \frac{K_{M}}{K_{M} + K_{R} + Q_{WG}}} \right\rbrack \times \left\lbrack \frac{V_{POST}}{K_{M} + K_{R}} \right\rbrack \left. \quad\left\lbrack {1 - \left( \frac{V_{POST}}{V_{PRE}} \right)^{{({K_{M} + K_{R}})}/Q_{WG}}} \right\rbrack \right\}}},} \right.}} & (R)\end{matrix}$

wherein K_(M) is a phosphorus mobilization clearance of the patient,Q_(WG) is a constant rate of fluid gain by the patient during theinterdialytic time interval (calculated byQ_(WG)=(t_(tx)Q_(UF))/(10080/F)), Q_(UF) is a constant rate of fluidremoved from the patient, V_(PRE) is a pre-dialysis distribution volumeof phosphorus of the patient prior to a hemodialysis treatment session,and V_(POST) is a post-dialysis distribution volume of phosphorus of thepatient at the end of a hemodialysis treatment session.

In any of the methods of determining G or C_(SS-PRE) in a hemodialysispatient when there is negligible net ultrafiltration or fluid removalfrom the patient during hemodialysis therapies and no weight gainbetween hemodialysis therapies, n C _(tx) and n C _(i) can be determinedusing the equations:

$\begin{matrix}{{n\; {\overset{\_}{C}}_{tx}} = {\frac{1}{t_{tx}}\left\{ {{\left\lbrack \frac{K_{M}t_{tx}}{K_{M} + K_{R} + K_{D}} \right\rbrack + {{\left\lbrack {1 - \frac{K_{M}}{K_{M} + K_{R} + K_{D}}} \right\rbrack \left\lbrack \frac{V_{PRE}}{K_{M} + K_{R} + K_{D}} \right\rbrack}\left. \quad\left\lbrack {1 - {\exp \left( {{- \left( {K_{D} + K_{R} + K_{M}} \right)}\; {t_{tx}/V_{PRE}}} \right)}} \right\rbrack \right\}}},{and}} \right.}} & \left( {{II}\text{-}S} \right) \\{{n\; {\overset{\_}{C}}_{i}} = {\frac{1}{{10080/F} - t_{tx}}\left\{ {{\left\lbrack \frac{K_{M}\left( {{10080/F} - t_{x}} \right)}{K_{M} + K_{R}} \right\rbrack + {\left\lbrack {\frac{K_{M}}{K_{M} + K_{R} + K_{D}} + {\left\lbrack {1 - \frac{K_{M}}{K_{M} + K_{R} + K_{D}}} \right\rbrack \times {\exp \left\lbrack {{- \left( {K_{D} + K_{R} + K_{M}} \right)}{t_{tx}/V_{PRE}}} \right\rbrack}} - \frac{K_{M}}{K_{M} + K_{R}}} \right\rbrack \times \left\lbrack \frac{V_{POST}}{K_{M} + K_{R}} \right\rbrack \left. \quad\left\lbrack {1 - {\exp \left( {{- \left( {K_{R} + K_{M}} \right)}{\left( {{10080/F} - t_{tx}} \right)/V_{POST}}} \right)}} \right\rbrack \right\}}},} \right.}} & \left( {{II}\text{-}T} \right)\end{matrix}$

It should be appreciated that the variables for equations II-S and II-Tcan be determined using any of the equations set forth herein.

In any of the methods of determining the G or the C_(SS-PRE) in ahemodialysis patient, K_(D) can be determined using the equation:

$\begin{matrix}{\mspace{79mu} {{K_{D} = {Q_{B}\frac{\left( {0.94 - {{Hct} \times 100}} \right)\left( {^{z} - 1} \right)}{\left( {^{z} - \frac{\left( {0.94 - {{Hct} \times 100}} \right)Q_{B}}{Q_{D}}} \right)}}}\mspace{79mu} {wherein}}} & \left( {{II}\text{-}U} \right) \\{\mspace{79mu} {{Z = {K_{o}A\frac{\left( {Q_{D} - {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B}}} \right)}{\left( {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B} \times Q_{D}} \right)}}},}} & \left( {{II}\text{-}V} \right) \\{{{K_{o}A} = {\frac{\left( {0.94 - {{Hct} \times 100}} \right)Q_{B,M} \times Q_{D,M}}{Q_{D,M} - {\left( {0.94\; - {{Hct} \times 100}} \right)Q_{B,M}}} \times {\ln\left( \frac{1 - {D_{M}/Q_{D,M}}}{1 - {D_{M}/\left\lbrack {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B,M}} \right\rbrack}} \right)}}},} & \left( {{II}\text{-}W} \right)\end{matrix}$

Q_(B) and Q_(D) are the blood and dialysate flow rates at which thedesired dialyzer clearance K_(D) is calculated using equations U and V.K_(O)A is a dialyzer mass transfer area coefficient for phosphateobtained as a result of a previous measurement where the set of bloodand dialysate flow rates Q_(B,M) and Q_(D,M) resulted in dialyzerclearance K_(D,M), and Hct is hematocrit count measured from patient'sblood sample. Alternatively, K_(D) can be determined at any time t usingthe equation:

$\begin{matrix}{K_{D} = \frac{{C_{D}\left( t_{s} \right)}{Q_{D}\left( t_{s} \right)}}{C\left( t_{s} \right)}} & \left( {{II}\text{-}X} \right)\end{matrix}$

wherein t_(s) is a sampling time and C_(D)(t_(s)) is a concentration ofphosphorus in a dialysate outflow at time t_(s) and Q_(D)(t_(s)) is adialysate flowrate at time t_(s) and C(t_(s)) is a serum phosphorusconcentration at time t_(s).

Alternative to non-linear least squares fitting, K_(M) can be determinedusing the following algebraic equation:

$\begin{matrix}{K_{M} = {C_{POST}\left( \frac{K_{D} - Q_{UF}}{C_{PRE} - C_{POST}} \right)}} & \left( {{II}\text{-}Y} \right)\end{matrix}$

wherein C_(POST) is a post-dialytic plasma phosphorus concentration, andC_(PRE) is a pre-dialysis plasma phosphorus concentration. G can bedetermined using the equation:

$\begin{matrix}{G = \frac{{I_{P}A_{P}} - {I_{B}P_{B}}}{10080}} & \left( {{II}\text{-}Z} \right)\end{matrix}$

wherein I_(P) is a weekly dietary intake of phosphorus of thehemodialysis patient, A_(P) is a percent phosphorus absorption of thehemodialysis patient, I_(B) is a weekly binder intake of thehemodialysis patient, and P_(B) is a binding power of the binder.

In an embodiment, K_(M) and V_(PRE) can be determined using the methodsof predicting phosphorus mobilization in a patient during hemodialysisas previously discussed. In this case, K_(M) and V_(PRE) are determinedby measuring C of the hemodialysis patient over a hemodialysis treatmentsession time and Q_(UF) calculated by a difference between pre- andpost-dialytic body weight of the patient during an initial hemodialysistreatment session divided by the total treatment time of the treatmentsession, and estimating K_(M) and V_(PRE) for the hemodialysis patientusing a non-linear least squares fitting to the governing transportequations having analytical solutions as follows:

$\begin{matrix}{{{C(t)} = {C_{PRE}\left\lbrack \frac{K_{M} + {\left( {K_{D} + K_{R} - Q_{UF}} \right)\left( \frac{V(t)}{V_{PRE}} \right)^{\frac{K_{M} + K_{D} + K_{R} - Q_{UF}}{Q_{UF}}}}}{K_{M} + K_{D} + K_{R} - Q_{UF}} \right\rbrack}}\mspace{79mu} {and}} & \left( {{II}\text{-}{AA}} \right) \\{\mspace{79mu} {{C(T)} = {C_{PRE} - {\left( {C_{PRE} - C_{POST}} \right)^{({- \frac{K_{M}T}{V_{PRE} - {Q_{UF}\; t_{tx}}}})}}}}} & \left( {{II}\text{-}{BB}} \right)\end{matrix}$

wherein t is a time during the hemodialysis treatment session, T is atime after an end of the hemodialysis treatment session, t_(tx) is atotal duration of the hemodialysis treatment session, C_(PRE) is apre-dialysis plasma phosphorus concentration, C_(POST) is apost-dialytic plasma phosphorus concentration, K_(M) is a phosphorusmobilization clearance of the patient, K_(R) is a residual renalclearance of phosphate, K_(D) is a dialyzer phosphate clearance, V_(PRE)is a pre-dialysis distribution volume of phosphorus of the patient, and

V(t)=V _(PRE) −Q _(UF) ×t  (II-CC).

The methods of determining G or C_(SS-PRE) in a hemodialysis patient canalso be used to determine or modify the appropriate treatments/dietarychanges to meet a desired phosphorus serum level in the hemodialysispatient over a period of time. For example, the methods can be used todetermine or modify a level of phosphorus intake so that C_(SS-PRE) ofthe hemodialysis patient ranges between about 3.6 mg/dl and 5.0 mg/dl.The methods can be used to determine or modify a phosphorus binderadministered to the patient so that C_(SS-PRE) of the hemodialysispatient ranges between about 3.6 mg/dl and 5.0 mg/dl. The methods canfurther be used to determine or modify an amount of phosphorus saltsupplements added to the dialysate so that C_(SS-PRE) of thehemodialysis patient ranges between about 3.6 mg/dl and 5.0 mg/dl.

The methods can be used to determine or modify the total hemodialysistreatment session time so that C_(SS-PRE) of the hemodialysis patientranges between about 3.6 mg/dl and 5.0 mg/dl. The methods can be used todetermine or modify the frequency F so that C_(SS-PRE) of thehemodialysis patient ranges between about 3.6 mg/dl and 5.0 mg/dl. Themethods can be used to determine or modify a required blood flowrateand/or a dialysate flowrate so that C_(SS-PRE) of the hemodialysispatient ranges between about 3.6 mg/dl and 5.0 mg/dl. It should beappreciated that the preferred range of C_(SS-PRE) can be patientspecific.

Specific steps of determining C_(SS-PRE) of a hemodialysis patient canbe performed using a computing device. Such a computing device caninclude a display device, an input device, a processor, and a memorydevice that stores a plurality of instructions, which when executed bythe processor, cause the processor to operate with the display deviceand the input device to: (a) receive data relating to G from at least adietary phosphorus intake of a hemodialysis patient or a urea kineticmodeling of the hemodialysis patient; (b) determine the C_(SS-PRE) ofthe patient using the equation:

$\begin{matrix}{C_{{SS} - {PRE}} = \frac{(G)\left( {10080/F} \right)}{{\left( {K_{D} + K_{R}} \right)n{\overset{\_}{C}}_{tx}t_{tx}} + {K_{R}n{{\overset{\_}{C}}_{i}\left( {{10080/F} - t_{tx}} \right)}}}} & \left( {{II}\text{-}{DD}} \right)\end{matrix}$

wherein F is a frequency of treatments per week, t_(tx) is a treatmenttime for one hemodialysis treatment session (e.g., in units of minutesper treatment session), K_(D) is a dialyzer phosphate clearance, K_(R)is a residual renal clearance of phosphate, n C _(tx) is the normalizedtime averaged plasma phosphorus concentration during a dialysistreatment, and n C _(i) is the normalized time averaged plasmaphosphorus concentration for an interdialytic interval; and (c) simulatethe effect of at least one of a patient parameter or a treatmentparameter on C_(SS-PRE) of the hemodialysis patient. It should beappreciated that the variables for equation II-DD can be determinedusing any of the appropriate equations set forth herein.

Another such computing device can include a display device, an inputdevice, a processor, and a memory device that stores a plurality ofinstructions, which when executed by the processor, cause the processorto operate with the display device and the input device to: (a)determine a net generation of phosphorus (“G”) using the equation:

$\begin{matrix}{G = {C_{{SS} - {PRE} - {IN}} \times \left\lbrack \frac{{\left( {K_{D} + K_{R}} \right)n{\overset{\_}{C}}_{tx}t_{tx}} + {K_{R}n{{\overset{\_}{C}}_{i}\left( {{10080/F} - t_{tx}} \right)}}}{10080/F} \right\rbrack}} & \left( {{II}\text{-}{EE}} \right)\end{matrix}$

wherein C_(SS-PRE-IN) is an initial, measured, steady state,pre-dialysis serum phosphorus level of the hemodialysis patient who ismaintained by a hemodialysis therapy (e.g., identified by K_(D), F, andt_(tx)) for a specified time prior to the calculation of G usingequation II-EE. The specified time can be, for example, at least oneweek, two weeks, three weeks, 1 month, 2 months, 3 months, 4 months ormore prior to time when G is calculated. F is a frequency of treatmentsper week, t_(tx) is a treatment time for one hemodialysis treatmentsession, K_(D) is a dialyzer phosphate clearance, K_(R) is a residualrenal clearance of phosphate, n C _(tx) is the normalized time averagedplasma phosphorus concentration during a dialysis treatment, and n C_(i) is the normalized time averaged plasma phosphorus concentration foran interdialytic interval; (b) predict steady state, pre-dialysis serumphosphorus levels (“C_(SS-PRE)”) of the hemodialysis patient using theequation:

$\begin{matrix}{{C_{{SS} - {PRE}} = \frac{(G)\left( {10080/F} \right)}{{\left( {K_{D} + K_{R}} \right)n{\overset{\_}{C}}_{tx}t_{tx}} + {K_{R}n{{\overset{\_}{C}}_{i}\left( {{10080/F} - t_{tx}} \right)}}}};} & \left( {{II}\text{-}{FF}} \right)\end{matrix}$

and

(c) simulate the effect of at least one of a patient parameter or atreatment parameter on C_(SS-PRE) of the hemodialysis patient. It shouldbe appreciated that the variables for equations II-EE and II-FF can bedetermined using any of the appropriate equations or methods set forthherein.

In any of the computing devices described herein, the information/dataobtained for the hemodialysis patient can be displayed/printed out andused by the healthcare provider to provide improved treatment andnutritional regimens for the hemodialysis patient. Any of the unknownfactors can be determined using any of the appropriate equations ormeasurements discussed herein for the methods of determining the steadystate, pre-dialysis serum phosphorus levels of a hemodialysis patient.

The computing devices can also be preprogrammed or run according tosoftware that causes the processor to operate with the display deviceand the input device to receive data relating to at least one of K_(R),K_(D), K_(M), V_(PRE), t_(tx), F, C_(PRE) about a month before ahemodialysis treatment session or a sampling time for collecting theserum phosphorus concentration. The computer device utilizes thisinformation to simulate the effect of one or more of these patientparameters or treatment parameters on C_(SS-PRE) of the hemodialysispatient, for example, using equations II-DD or II-FF (e.g., seeing how achange in one or more of the patient parameters or treatment parametersimpacts C_(SS-PRE)). The computing device can be preprogrammed todisplay a treatment regimen of the hemodialysis patient so thatC_(SS-PRE) is within a desired range using any of the methods disclosedherein. In an embodiment, the computing device can be system 10described in section I.

Any of the computer devices described herein (including any portions ofsystem 10 described in section I) can be a device having a processorcapable of receiving data and performing calculations based on thatdata. Such computing device can be, for example, a handheld clientdevice, personal computer client device, database server, etc.). A moredetailed block diagram of the electrical systems of the computingdevices described herein is illustrated in FIG. 10. Although theelectrical systems of these computing devices may be similar, thestructural differences between these devices are well known. Forexample, a typical handheld client device is small and lightweightcompared to a typical database server.

In FIG. 10, an example computing device 202 preferably includes one ormore processors 204 electrically coupled by an address/data bus 206 toone or more memory devices 208, other computer circuitry 210, and one ormore interface circuits 212. Processor 204 may be any suitableprocessor, such as a microprocessor from the INTEL PENTIUM® family ofmicroprocessors. Memory 208 preferably includes volatile memory andnon-volatile memory. Preferably, memory 208 stores a software program(e.g., Matlab, C++, Fortran, etc.) that can perform the calculationsnecessary according to embodiments described herein and/or thatinteracts with the other devices in a hemodialysis system. This programmay be executed by processor 204 in any suitable manner. Memory 208 mayalso store digital data indicative of documents, files, programs, webpages, etc. retrieved from another computing device and/or loaded via aninput device 214.

Interface circuit 212 may be implemented using any suitable interfacestandard, such as an Ethernet interface and/or a Universal Serial Bus(“USB”) interface. One or more input devices 214 may be connected to theinterface circuit 212 for entering data and commands into computingdevice 202. For example, input device 214 may be a keyboard, mouse,touch screen, track pad, track ball, isopoint, and/or a voicerecognition system.

One or more displays, printers, speakers, and/or other output devices216 may also be connected to computing device 202 via interface circuit212. Display 216 may be a cathode ray tube (“CRT”), a liquid crystaldisplay (“LCD”), or any other type of display. Display 216 generatesvisual displays of data generated during operation of computing device202. The visual displays may include prompts for human input, run timestatistics, measured values, calculated values, data, etc.

One or more storage devices 218 may also be connected to computingdevice 202 via interface circuit 212. For example, a hard drive, CDdrive, DVD drive, and/or other storage devices may be connected tocomputing device 202. Storage devices 218 may store any type of suitabledata.

Computing device 202 may also exchange data with other network devices220 via a connection to a network 230. The network connection may be anytype of network connection, such as an Ethernet connection, digitalsubscriber line (“DSL”), telephone line, coaxial cable, etc. This allowscomputing device 202 to communicate with a suitable dialysis machine, apatient database and/or a hospital network depending on the desiredapplications.

EXAMPLES

By way of example and not limitation, the following examples areillustrative of various embodiments of the present disclosure andfurther illustrate experimental testing conducted with the systems andmethods in accordance with embodiments of the present disclosure.

Example 1 Objective

The objective of this analysis was to demonstrate the non-linear leastsquares fitting procedure for estimating patient specific parameters(e.g., K_(M) and V_(PRE)) from a pseudo one-compartment model usingclinical data, and to evaluate the validity of parameter estimates overdifferent HD treatment modalities.

Pseudo One-Compartment Model

A conceptual description of the pseudo one-compartment model is shown inFIG. 11. In this model, phosphorus is removed by the dialyzer from acompartment of volume V, also called the distribution volume, andphosphorus concentration C. The distribution volume is assumed to be inequilibrium with plasma. Phosphorus mobilization into this compartmentoccurs from phosphorus pools in the body that are inaccessible to thedialyzer. These pools are represented as a large compartment with aconstant phosphorus concentration equal to the pre-dialytic plasmaphosphorus concentration (“C_(PRE)”). The rate of phosphorusmobilization into the distribution volume is described as the differencebetween pre-dialytic and instantaneous plasma phosphorus levelsmultiplied by the phosphorus mobilization clearance (“K_(M)”). K_(M) isanalogous to an intercompartmental mass transfer coefficient, and isassumed to be constant during the treatment and the post-dialyticrebound periods. Residual renal clearance of phosphate is neglected inthis example.

Changes in the volume and phosphorus concentration of phosphorusdistribution volume during and shortly after an HD treatment session arerepresented by equations E-A1 and E-A2,

$\begin{matrix}{\frac{({VC})}{t} = {{K_{M}\left( {C_{PRE} - C} \right)} - {\Theta \; K_{D}C}}} & {E\text{-}A\; 1} \\{\frac{(V)}{t} = {{- \Theta}\; Q_{UF}}} & {E\text{-}A\; 2}\end{matrix}$

where, Θ is a variable that indicates whether dialysis treatment istaking place (Θ=1) or not (Θ=0), K_(D) is the dialyzer phosphateclearance, and Q_(UF) is the ultrafiltration (“UF”) rate. The kineticmodel described above also assumes that all fluid removed during thetreatment is from the distribution volume of phosphorus.

Closed form analytical solutions to the time-dependent plasma phosphorusconcentration can be obtained by integrating equations 1 and 2. For theintradialytic (Θ=1) and rebound (Θ=0) periods, the time dependence ofphosphorus concentration can be expressed as shown by equations E-A3 andE-A4 respectively:

$\begin{matrix}{{C(t)} = {C_{PRE}\left\lbrack \frac{K_{M} + {\left( {K_{D} - Q_{UF}} \right)\left( \frac{V_{PRE} - {Q_{UF} \times t}}{V_{PRE}} \right)^{\frac{K_{M} + K_{D} - Q_{UF}}{Q_{UF}}}}}{K_{M} + K_{D} - Q_{UF}} \right\rbrack}} & {E\text{-}A\; 3} \\{\mspace{79mu} {{C(T)} = {C_{PRE} - {\left( {C_{PRE} - C_{POST}} \right)^{({- \frac{K_{M}T}{V_{PRE} - {Q_{UF}t_{tx}}}})}}}}} & {E\text{-}A\; 4}\end{matrix}$

wherein V_(PRE) is the pre-dialytic distribution volume of phosphorus, ttime during the treatment, T is time after the end of the treatment, andt_(tx) is total duration of treatment prior to the rebound period. It isalso assumed that distribution volume of phosphorus remains constantduring the post-dialytic rebound period.

Methods

Clinical data was obtained from 5 chronic hemodialysis patients whoparticipated in a crossover trial. Patients underwent a short HD (“SHD”)treatment session and a conventional HD (“CHD”) treatment session oneweek apart. Blood samples were collected at t=0, 60, 90 min during SHDand, t=0, 30, 60, 120, 180 min during CHD treatments. Dialysate sampleswere collected 60 min after the start of treatments to determinedialyzer phosphate clearance. Additional blood samples were collected att=10 seconds, 2, 10, 30, 60 min after the end of treatments. Plasma anddialysate samples were assayed for phosphorus.

Patient specific parameters (K_(M) and V_(PRE)) were estimated bynon-linear least squares fitting to clinical data using equations 3 and4. Least squares fitting was performed using a scientific computationalsoftware (MATLAB v2008a, Mathworks, Natick, Mass., USA). The model wasfit to SHD and CHD data separately, resulting in two sets of K_(M) andV_(PRE) estimates for each patient. Q_(UF) was calculated by thedifference between pre-dialytic and post-dialytic body weight of thepatient divided by total treatment time. Dialyzer phosphate clearancewas calculated according to equation E-A5, where C_(D) is theconcentration of phosphorus in the dialysate outflow and Q_(D) is thedialysate flow rate.

$\begin{matrix}{K_{D} = \frac{{C_{D}\left( {t = {60\mspace{14mu} \min}} \right)} \times Q_{D}}{C\left( {t = {60\mspace{14mu} \min}} \right)}} & {E\text{-}A\; 5}\end{matrix}$

Non-linear regression fits to clinical data from each patient arepresented in FIGS. 12-16. FIG. 12 shows modeled and measured plasmaphosphorus concentrations for patient 1 during SHD and CHD treatmentsessions. FIG. 13 shows modeled and measured plasma phosphorusconcentrations for patient 2 during SHD and CHD treatment sessions. FIG.14 shows modeled and measured plasma phosphorus concentrations forpatient 3 during SHD and CHD treatment sessions. FIG. 15 shows modeledand measured plasma phosphorus concentrations for patient 4 during SHDand CHD treatment sessions. FIG. 16 shows modeled and measured plasmaphosphorus concentrations for patient 5 during SHD and CHD treatmentsessions. There was good agreement between modeled and measured plasmaphosphorus concentration during SHD and CHD treatment sessions.

A detailed summary of parameter estimates are presented in Table II.1.Parameter estimates varied considerably among patients, but for eachpatient, estimates obtained from SHD and CHD treatment sessions weresimilar. Low values for standard errors (“SE”) indicate high precisionin parameter estimates.

These results suggest that K_(M) and V_(PRE) are patient specificparameters independent of HD treatment time. Therefore extending kineticmodel predictions from thrice-weekly conventional (3-4 hours) toshort-daily (2-3 hours) and nocturnal HD (6-10 hours) therapies may befeasible using K_(M) and V_(PRE) values estimated from conventional HDtreatments.

TABLE II.1 Estimated values of patient parameters from SHD and CHDtreatment sessions. K_(M) (SHD) K_(M) (CHD) V_(PRE) (SHD) V_(PRE) (CHD)Patient ID (ml/min) (ml/min) (L) (L) 1  66 ± 10 55 ± 5 11.2 ± 1.4  11.8± 1.0 2 78 ± 5 84 ± 6 8.1 ± 0.5  9.6 ± 0.7 3 67 ± 8  96 ± 12 14.6 ± 1.3 14.0 ± 2.2 4 104 ± 18 101 ± 11 7.3 ± 1.1  8.9 ± 0.9 5 58 ± 5 50 ± 5 9.0± 0.7 10.7 ± 1.1 Parameter estimates are expressed as estimated value ±standard error

Example 2 Objective

The objective of this study was to demonstrate the application of asimple method for estimating the patient parameter K_(M) from a pseudoone-compartment model using data from conventional 4-hour hemodialysistreatments, and to evaluate the accuracy of estimated K_(M) values viacomparison to results obtained using non-linear least squares fitting.

Methods

Clinical data was obtained from 5 chronic hemodialysis patients whounderwent CHD treatments. Blood samples were collected at t=0, 30, 60,120, 180 min during the treatments and 10 seconds, 2, 10, 30, 60 minafter the end of treatments. Dialysate samples were collected 60 minfrom the start of treatments to determine dialyzer phosphate clearance.Plasma and dialysate samples were assayed for phosphorus.

K_(M) was computed for each patient using equation E-B1, where C_(POST)is the post-dialytic plasma phosphorus concentration, K_(D) is thedialyzer phosphate clearance, Q_(UF) is the ultrafiltration rate or netfluid removal rate, and C_(PRE) is the pre-dialytic plasma phosphorusconcentration.

$\begin{matrix}{K_{M} = {C_{POST}\left( \frac{K_{D} - Q_{UF}}{C_{PRE} - C_{POST}} \right)}} & {E\text{-}B\; 1}\end{matrix}$

Q_(UF) was calculated by the difference between pre-dialytic andpost-dialytic body weight of the patient divided by total treatmenttime. Dialyzer phosphate clearance was calculated according to equationE-B2, where C_(D) is concentration of phosphorus in the dialysateoutflow and Q_(D) is the dialysate flow rate.

$\begin{matrix}{K_{D} = \frac{{C_{D}\left( {t = {60\mspace{14mu} \min}} \right)} \times Q_{D}}{C\left( {t = {60\mspace{14mu} \min}} \right)}} & {E\text{-}B\; 2}\end{matrix}$

To evaluate the accuracy of equation E-B1, computed K_(M) values werecompared with estimates obtained using non-linear least squares fittingto 10 measured intradialytic and post-dialytic rebound concentrations ofplasma phosphorus, as described in Example 1.

Results

K_(M) values for individual patients, computed using equation E-B1 andestimated from non-linear least squares fitting to frequent measurementsare presented in Table 11.2 along with the pre-dialytic andpost-dialytic plasma phosphorus concentrations, ultrafiltration rate,and the dialyzer phosphate clearance. There was good agreement betweenK_(M) values obtained using equation E-B1, and non-linear least squaresfitting.

These results suggest that equation E-B1 can be used as an alternativeto performing non-linear least squares fitting to frequent measurementsof plasma phosphorus concentrations for the estimation of patientspecific K_(M). Its simple algebraic form and utilization of onlypre-dialysis and post-dialysis blood samples makes it a practical methodto study kinetics of phosphorus mobilization during HD treatments on anindividual patient basis.

TABLE II.2 K_(M) values for individual patients computed from equationE-B1, and estimated using non-linear least squares fitting (“NLSQ”)K_(D) K_(M) Patient C_(PRE) C_(POST) Q_(UF) (ml/ K_(M) (E-B1) (NLSQ) ID(mg/dl) (mg/dl) (ml/min) min) (ml/min) (ml/min) 1 8.4 2.3 8 154 55 56 24.4 1.8 7 131 85 84 3 6.7 3.2 8 129 110 96 4 7.3 3.2 12 135 96 102 5 4.21.5 12 117 58 51 Abbreviations: SD, standard deviation

Example 3 Steady State Phosphorus Mass Balance Model Objectives

As previously discussed, the inventors have proposed a kinetic model fordescribing changes in serum or plasma phosphorus concentrations duringhemodialysis (more generally extracorporeal treatments) and thepost-dialytic rebound period. That kinetic model allows prediction ofintradialytic phosphorus concentrations as a function of time and totalphosphate removal from the knowledge of: 1) the pre-dialysisconcentration of phosphorus in plasma or serum, 2) the dialyzerclearance of phosphate, 3) the volume of distribution of phosphorus, 4)the amount of fluid removed during the treatment, and 5) apatient-specific phosphorus mobilization clearance. The steady statephosphorus mass balance model described below will be used incombination with the previous kinetic model to allow determination ofpre-dialysis serum phosphorus concentration for individual patientsunder any hemodialysis treatment prescription when the above parameters2-5 are established, the frequency of hemodialysis treatments per weekand the hemodialysis treatment duration are prescribed, and the netgeneration of phosphorus (defined below), and residual kidney or renalphosphorus clearance are all known. Alternatively, the steady statephosphorus mass balance model in combination with the previous kineticmodel can be used to determine the net generation of phosphorus for agiven patient when the above parameters 1-5 are established and thefrequency of hemodialysis treatments per week, the hemodialysistreatment duration and the residual kidney phosphorus clearance areknown. As in other mass balance models, the patient is assumed to be insteady state.

Steady State Mass Balance Model

The model used to describe steady state phosphorus mass balance over atime averaged period, i.e., a week, for a patient treated byhemodialysis is shown schematically in FIG. 17. This model is ageneralized version of the kinetic model described previouslycharacterizing phosphorus kinetics during treatments and thepost-dialytic rebound period. The model assumes that phosphorus isdistributed in a single, well-mixed compartment.

There are several pathways that result in changes in the concentrationof phosphorus (“C”) within its distribution volume (“V”). Dietary intakeof phosphorus is derived mostly from dietary protein; however, foodadditives can also contain significant amounts of phosphate. The amountof dietary phosphorus intake often exceeds the amount of phosphate thatcan be removed by conventional thrice weekly hemodialysis; therefore,dialysis patients are frequently prescribed oral phosphate binders tocontrol serum phosphorus concentrations. Dietary intake of phosphorusminus the amount of phosphate that is bound and not absorbedintestinally can be combined and is defined as the net generation ofphosphorus (“G”); this parameter will be assumed to be a constant inthis model. Phosphorus can be removed directly from its distributionvolume by dialyzer clearance (“K_(D)”) or residual renal or kidneyclearance (“K_(R)”). As described previously in the kinetic model,phosphorus can also be mobilized from other compartments at a rateproportional to the difference between the instantaneous and thepre-dialysis (“C_(PRE)”) phosphorus concentrations. The proportionalityparameter has been termed the phosphorus mobilization clearance(“K_(M)”). Phosphorus can also be deposited into tissues depending on acritical tissue concentration (“C_(t)”); this process has been modeledas a tissue deposition clearance (“K_(r)”).

The model shown in FIG. 17 was devised to identify all major pathwaysfor phosphorus distribution in hemodialysis patients; however, it islikely too complex to be useful clinically and requires simplification.Specifically, tissue deposition of phosphorus is likely to be smallcompared with the other pathways and can be neglected as a firstapproximation in this mass balance model.

A mass balance differential equation for the model shown in FIG. 17(neglecting tissue deposition of phosphorus) is the following:

$\begin{matrix}{\frac{({CV})}{t} = {G - {K_{D}C} - {K_{R}C} + {K_{M}\left( {C_{PRE} - C} \right)}}} & {E\text{-}C\; 1}\end{matrix}$

Assuming that hemodialysis treatment sessions are symmetrically placedthroughout the week, integrating equation E-C1 over a week is equivalentto integrating it over one complete cycle of a treatment (with atreatment time of t_(tx)) and an interdialytic interval betweentreatments (with a time of T_(i)). Note that lower case t indicates timeduring the treatment and it varies from 0 to t_(tx), whereas T indicatesthe time during the interdialytic interval and it varies from 0 toT_(i). The values of t_(tx) and T_(i) are related and dependent on thenumber of treatments per week (the above mathematical analysis isgeneral and applies to an arbitrary number of hemodialysis treatmentsessions per week; here, F denotes the number of treatments per week).If t and T are reported in units of hours, then T_(i)=168/F−t_(tx). If tand T are reported in units of minutes, then T_(i)=10080/F−t_(tx). Thisintegration results in equation E-C2 after some rearrangement.

$\begin{matrix}{{\Delta ({CV})} = {{K_{M}{\int_{0}^{t_{tx} + T_{i}}{\left\lbrack {C_{PRE} - {C(\tau)}} \right\rbrack \ {\tau}}}} = {{G\left( {t_{tx} + T_{i}} \right)} - {\left( {K_{D} + K_{R}} \right){\int_{t = 0}^{t_{tx}}{{C(\tau)}\ {\tau}}}} - {K_{R}{\int_{T = 0}^{T_{i}}{{C(\tau)}\ {\tau}}}}}}} & {E\text{-}C\; 2}\end{matrix}$

where Δ(CV) indicates the change in phosphorus mass within its volume ofdistribution. Note that the second term on the left hand side of thisequation indicates (minus) the mass of phosphorus transported into V viathe mobilization pathway.

Assuming further that the patient is in steady state and totalphosphorus mass in the body (i.e., left hand side of equation E-C2) doesnot change during one complete cycle of a treatment and an interdialyticinterval, the left hand side of equation E-C2 must be zero; thus,equation E-C2 is reduced in steady state to the following:

$\begin{matrix}{0 = {{G\left( {t_{tx} + T_{i}} \right)} - {\left( {K_{D} + K_{R}} \right){\int_{t = 0}^{t_{tx}}{{C(\tau)}\ {\tau}}}} - {K_{R}{\int_{T = 0}^{T_{i}}{{C(\tau)}\ {\tau}}}}}} & {E\text{-}C\; 3}\end{matrix}$

To use this integrated mass balance equation, it is necessary tocalculate both the integrals in equation E-C3. Note that the firstintegral is over the time period during the treatment and the second isover the time period during the interdialytic interval.

To evaluate the integrals in equation E-C3, the inventors make twoadditional assumptions. First, the inventors assume that changes inphosphorus concentration during a treatment and the post-dialyticrebound period can be described by the previously proposed kinetic modelwhere net generation of phosphorus can be neglected. Second, theinventors assume that this same kinetic model describes changes inphosphorus concentration during the entire interdialytic interval, notjust the post-dialysis rebound period. The equation describing thekinetic model is the following

$\begin{matrix}{\frac{({CV})}{t} = {{{- K_{D}}C} - {K_{R}C} + {K_{M}\left( {C_{PRE} - C} \right)}}} & {E\text{-}C\; 4}\end{matrix}$

Equation E-C4 can be solved analytically; the time dependence of theserum phosphorus concentration during the treatment can be described by

$\begin{matrix}{\frac{C(t)}{C_{PRE}} = {\frac{K_{M}}{K_{M} + K_{R} + K_{D} - Q_{UF}} + {\left\lbrack {1 - \frac{K_{M}}{K_{M} + K_{R} + K_{D} - Q_{UF}}} \right\rbrack \left\lbrack \frac{V(t)}{V_{PRE}} \right\rbrack}^{{({K_{M} + K_{D} + K_{R} - Q_{UF}})}/Q_{UF}}}} & {E\text{-}C\; 5}\end{matrix}$

where it has been assumed that fluid is removed from the patient at aconstant rate (“Q_(UF)”), such that the distribution volume decreaseslinearly from its initial, pre-dialysis value (“V_(PRE)”) throughout thetreatment. Stated in mathematical terms,

V(t)=V _(PRE) −Q _(UF) ×t  E-C6

Thus, it is assumed that all fluid removed during the treatment isremoved from the distribution volume of phosphorus.

During the rebound period (and the entire interdialytic interval),equation E-C4 remains valid except that K_(D) is zero. Assuming that thepatient gains fluid at a constant rate (“Q_(WG)”) during theinterdialytic interval, such that the distribution volume increaseslinearly from its initial post-dialysis value (“V_(POST)”), theanalytical solution describing the time dependence of the serumphosphorus concentration during the interdialytic interval is

$\begin{matrix}{\frac{C(T)}{C_{PRE}} = {{\left\lbrack {\frac{C_{POST}}{C_{PRE}} - \frac{K_{M}}{K_{M} + K_{R} + Q_{WG}}} \right\rbrack \left\lbrack \frac{V(T)}{V_{POST}} \right\rbrack}^{{- {({K_{M} + K_{R} + Q_{WG}})}}/Q_{WG}} + \frac{K_{M}}{K_{M} + K_{R} + Q_{WG}}}} & {E\text{-}C\; 7}\end{matrix}$

where the time dependence of the distribution volume during theinterdialytic interval is described by

V(T)=V(T=0)+Q _(WG) ×T=V _(POST) +Q _(WG) ×T=V _(PRE) −Q _(UF) ×t _(tx)+Q _(WG) ×T  E-C8

Note that it is assumed that all fluid gained during the interdialyticinterval is confined to the distribution volume of phosphorus. The twointegrals in equation E-C3 can now be obtained by integrating equationsE-C5 and E-C7. The normalized time averaged plasma phosphorusconcentration during dialysis treatments (n C _(tx)) is obtained byintegrating equation E-C5:

${n{\overset{\_}{C}}_{tx}} = {{\left( \frac{1}{t_{tx}C_{PRE}} \right){\int_{0}^{t_{tx}}{{C(\tau)}\ {\tau}}}} = {\frac{1}{t_{tx}}\left\{ {\left\lbrack \frac{K_{M}t_{tx}}{K_{M} + K_{R} + K_{D} - Q_{UF}} \right\rbrack + {\left\lbrack {1 - \frac{K_{M}}{K_{M} + K_{R} + K_{D} - Q_{UF}}} \right\rbrack \left. \quad{\left\lbrack \frac{V_{PRE}}{K_{M} + K_{R} + K_{D}} \right\rbrack \left\lbrack {1 - \left( \frac{V_{POST}}{V_{PRE}} \right)^{{({K_{M} + K_{R} + K_{D}})}/Q_{UF}}} \right\rbrack} \right\}}} \right.}}$

Integration of equation E-C7 and calculation of C_(POST)/C_(PRE) fromequation E-C5 when t=t_(tx) gives the normalized time averaged plasmaphosphorus concentration for the interdialytic interval (n C _(i)):

$\begin{matrix}{\frac{C(T)}{C_{PRE}} = {{\left\lbrack {\frac{C_{POST}}{C_{PRE}} - \frac{K_{M}}{K_{M} + K_{R} + Q_{WG}}} \right\rbrack \left\lbrack \frac{V(T)}{V_{POST}} \right\rbrack}^{{- {({K_{M} + K_{R} + Q_{WG}})}}/Q_{WG}} + \frac{K_{M}}{K_{M} + K_{R} + Q_{WG}}}} & {E - {C7}}\end{matrix}$

When equations E-C3, E-C9 and E-C10 are combined, the resulting equationgoverning phosphorus mass balance at steady state can be expressed as:

$\begin{matrix}{G = {C_{{PRE} - {SS} - {IN}} \times \frac{{\left( {K_{D} + K_{R}} \right)n{\overset{\_}{C}}_{tx}t_{tx}} + {K_{R}n{{\overset{\_}{C}}_{i}\left( {{10080/F} - t_{tx}} \right)}}}{10080/F}}} & {E - {C11}}\end{matrix}$

C_(SS-PRE-IN) is an initial, measured, steady state, pre-dialysis serumphosphorus level of the hemodialysis patient who is maintained by ahemodialysis therapy (e.g., identified by K_(D), F and t_(tx)) for aspecified time prior to the calculation of G using equation E-C11.Equation E-C11 can be used to predict G if the pre-dialysis serumconcentration is measured in a patient with knowledge of varioustreatment and patient parameters.

Once G has been calculated using equation E-C11 or estimated by othermethods, it can be used in predicting the effect of changes inhemodialysis treatment parameters on the steady state serum phosphorusconcentration by rearranging equation E-C11 to the following:

$\begin{matrix}{C_{{PRE} - {SS}} = {G \times \frac{10080/F}{{\left( {K_{D} + K_{R}} \right)n{\overset{\_}{C}}_{tx}t_{tx}} + {K_{R}n{{\overset{\_}{C}}_{i}\left( {{10080/F} - t_{tx}} \right)}}}}} & {E - {C12}}\end{matrix}$

In general, there is an optimal range of pre-dialysis serum phosphoruslevels in patients with end stage renal disease; thus, equation E-C12can be used to optimize the prescription to obtain a desiredpre-dialysis serum phosphorus concentration. It should be mentioned thatchanges in the hemodialysis prescription or in patient behavior (e.g.,dietary intake) could lead to changes in G; thus, the iterative use ofequations E-C11 and E-C12 to optimize C_(SS-PRE) may be necessary.

Equation E-C12 can be used to predict the pre-dialysis serum phosphorusconcentration with knowledge of various treatment and patientparameters. Thus, equations E-C9 to E-C12 can be viewed as defining asteady state phosphorus mass balance of the hemodialysis patient.

Equations E-C5 to E-C10 do not apply when there is negligible netultrafiltration or fluid removal from the patient during the treatmentand no weight gain between treatments. When there is negligibleultrafiltration during the treatment and no weight gain betweentreatments, equations E-C5 to E-C10 become:

$\begin{matrix}{\frac{C(t)}{C_{PRE}} = {\frac{K_{M}}{K_{M} + K_{R} + K_{D}} + {\left\lbrack {1 - \frac{K_{M}}{K_{M} + K_{R} + K_{D}}} \right\rbrack \times {\exp \left\lbrack {{- \left( {K_{D} + K_{R} + K_{M}} \right)}{t/V_{PRE}}} \right\rbrack}}}} & {E - {C5A}} \\{{V(t)} = {{V\left( {t = 0} \right)} = V_{PRE}}} & {E - {C6A}} \\{\frac{C(T)}{C_{PRE}} = {{\left\lbrack {\frac{C_{POST}}{C_{PRE}} - \frac{K_{M}}{K_{M} + K_{R}}} \right\rbrack \times {\exp \left\lbrack {{- \left( {K_{R} + K_{M}} \right)}{T/V_{POST}}} \right\rbrack}} + \frac{K_{M}}{K_{M} + K_{R}}}} & {E - {C7A}} \\{{V(T)} = {{V\left( {T = 0} \right)} = {V_{POST} = V_{PRE}}}} & {E - {C8A}} \\{{n{\overset{\_}{C}}_{tx}} = {{\left( \frac{1}{t_{tx}C_{PRE}} \right){\int_{0}^{t_{tx}}{{C(\tau)}\ {\tau}}}} = {\frac{1}{t_{tx}}\begin{Bmatrix}{\left\lbrack \frac{K_{M}t_{tx}}{K_{M} + K_{R} + K_{D}} \right\rbrack +} \\{{\begin{bmatrix}{1 -} \\\frac{K_{M}}{K_{M} + K_{R} + K_{D}}\end{bmatrix}\left\lbrack \frac{V_{PRE}}{K_{M} + K_{R} + K_{D}} \right\rbrack}\begin{bmatrix}{1 -} \\{\exp \left( {{- \left( {K_{D} + K_{R} + K_{M}} \right)}{t_{tx}/V_{PRE}}} \right)}\end{bmatrix}}\end{Bmatrix}}}} & {E - {C9A}} \\{{n{\overset{\_}{C}}_{i}} = {{\left( \frac{1}{T_{i}C_{PRE}} \right){\int_{0}^{T_{i}}{{C(\tau)}\ {\tau}}}} = {\frac{1}{{10080/F} - t_{tx}}\begin{Bmatrix}{\left\lbrack \frac{K_{M}\left( {{10080/F} - t_{tx}} \right)}{K_{M} + K_{R}} \right\rbrack +} \\{\begin{bmatrix}{\frac{K_{M}}{K_{M} + K_{R} + K_{D}} +} \\{\left\lbrack {1 - \frac{K_{M}}{K_{M} + K_{R} + K_{D}}} \right\rbrack \times} \\{{\exp \left\lbrack {{- \left( {K_{D} + K_{R} + K_{M}} \right)}{t_{tx}/V_{PRE}}} \right\rbrack} -} \\\frac{K_{M}}{K_{M} + K_{R}}\end{bmatrix} \times} \\{\left\lbrack \frac{V_{POST}}{K_{M} + K_{R}} \right\rbrack \left\lbrack {1 - {\exp \left( {{- \left( {K_{R} + K_{M}} \right)}{\left( {{10080/F} - t_{tx}} \right)/V_{POST}}} \right)}} \right\rbrack}\end{Bmatrix}}}} & {E - {C10A}}\end{matrix}$

Under these conditions, equations E-C11 and E-C12 can be used with thesemodified equations.

Applications

Equations E-C9 to E-C11 can be used calculate patient specific values ofG from the data analyzed in Examples 1 and 2. The measured pre-dialysisconcentrations of phosphorus (“C_(PRE)”) and the calculated values of Gfrom that data during conventional 4-hour treatments are summarized inthe Table II.3 below.

TABLE II.3 Patient Label C_(PRE) (mg/dl) G (g/week) Patient 1 8.4 4.13Patient 2 4.4 1.71 Patient 3 6.7 3.64 Patient 4 7.3 3.83 Patient 5 4.21.68

The calculated G values are consistent with expected phosphorus netgeneration rates in chronic hemodialysis patients.

Equations E-C9 to E-C12 can also be used to simulate the effect ofpatient parameters (G, K_(M) and K_(R)) and treatment parameters(t_(tx), K_(D) and F) on pre-dialysis serum phosphorus concentrations.Several different simulations will be illustrated; K_(R) will be assumedto be zero in these simulations. These simulation examples show that thesteady state mass balance model predicts results that are similar tothose expected from the medical literature.

The importance of treatment time under conditions relevant to thriceweekly hemodialysis is of high clinical interest; therefore, theinventors examined the effect of treatment time on the pre-dialysisserum phosphorus concentration at the same dialysis dose or urea Kt/V.The inventors used the above described model to perform computersimulations of steady state serum phosphorus concentrations duringthrice weekly hemodialysis. Simulations were performed for a fixed netphosphorus intake or generation rate (dietary intake minus absorption byoral binders), a urea Kt/V of 1.4 and a constant relationship betweendialyzer clearances of phosphate and urea (i.e., the inventors assumedthat dialyzer clearance of phosphate was one-half of that for urea andthat the phosphorus distribution volume was one-third of that for urea).

Simulated pre-dialysis serum phosphorus concentrations (in mg/dl) aretabulated in Table II.4 below for hypothetical patients with differentK_(M), a post-dialysis phosphorus distribution volume of 12 L, and netfluid removal per treatment of 2 L.

TABLE II.4 Treatment Time K_(M) (ml/min) (min) 50 100 200 180 7.65 6.916.07 240 7.36 6.56 5.75 300 7.07 6.24 5.47

For a given patient, increasing treatment time at a given urea Kt/Vresulted in modest decreases in pre-dialysis serum phosphorusconcentration. Similar findings were obtained for other values of ureaKt/V between 1.0 and 2.0 (results not shown). These predictions showthat use of urea Kt/V as the only measure of dialysis dose or dialysisadequacy does not account for differences in phosphate removal.

FIG. 18 illustrates the effect of treatment frequency per se onpre-dialysis serum phosphorus concentration as a function of dialyzerphosphate clearance where it was assumed that K_(M) was equal to 100ml/min, V was assumed equal to 10 L with no fluid removal during thetreatment, the treatment time was 630 minutes/week and the netgeneration of phosphorus was kept constant at 3 g/week. There is arelatively uniform decrease in pre-dialysis serum phosphorusconcentration upon increasing treatment frequency from 3-times per weekto 6-times per week, independent of dialyzer phosphate clearance. Theuniformity of the decrease was surprising, it varied between 0.98 and1.00 when K_(M)=100 ml/min as shown in this figure (for dialyzerphosphate clearance between 100 and 200 ml/min). Relatively uniformdecreases in concentration were also evident for K_(M) of 50, 150 and200 ml/min (data not shown). The respective decreases in pre-dialysisserum phosphorus concentration were 1.67-1.84 (K_(M)=50 ml/min),0.60-0.63 (K_(M)=150 ml/min) and 0.40-0.43 (K_(M)=200 ml/min).

FIG. 19 illustrates the effects of an increase in treatment time andfrequency with reference to nocturnal forms of hemodialysis. Doublingtreatment time during three-times per week therapy produces substantialreductions in pre-dialysis serum phosphorus concentration. When thesereductions are compared with those for doubling treatment frequency atthe same weekly treatment time as in FIG. 18, it can be concluded thatdoubling treatment time (at the same treatment frequency) has a moresubstantial effect on pre-dialysis serum phosphorus concentrations thandoes doubling frequency (at the same weekly treatment time). Doublingboth treatment time and treatment frequency reduces pre-dialysis serumphosphorus concentration even further.

Previous clinical studies have shown that patients treated by shortdaily hemodialysis often have a higher rate of protein catabolism (orprotein nitrogen appearance) and a higher dietary intake of both proteinand phosphorus. The increase in the protein nitrogen appearance rate hasbeen reported to be approximately 20%. Therefore, the inventorsevaluated the effect of increasing treatment frequency and treatmenttime relevant to short daily hemodialysis on pre-dialysis serumphosphorus concentration when net generation of phosphorus was increasedup to 30% more than during conventional 3-times per week hemodialysistherapy. These results assuming post-dialytic phosphorus distributionvolume of 10 L with 6 L of fluid removal per week are shown in FIGS.20-21 for K_(M)=50 ml/min and FIGS. 22-23 for K_(M)=150 ml/min. Asexpected, pre-dialysis serum phosphorus concentrations were higher forlower K_(M) values. The interactions between K_(M), dialyzer phosphateclearance (“K_(D)”) and treatment frequency and treatment time arecomplex when net generation of phosphorus is increased. Some specificvalues of pre-dialysis serum phosphorus concentration in mg/dl duringconventional 3-times per week hemodialysis (“CHD”) and during shortdaily hemodialysis (“SDHD”) when net generation of phosphorus during thelatter therapy was increased by 20% are tabulated in Table II.5.

TABLE II.5 K_(M) = 50 ml/min K_(M) = 150 ml/min CHD SDHD SDHD CHD SDHDSDHD Treatment 240 120 180 240 120 180 Time (min) K_(D) 80 8.34 8.536.38 6.98 7.85 5.52 (ml/ 110 7.11 6.97 5.36 5.62 6.22 4.42 min) 140 6.436.10 4.80 4.86 5.29 3.80

Increasing hemodialysis treatment session frequency without an increasein weekly treatment time (CHD to SDHD at a treatment time of 120 min)may result in either an increase or a decrease in pre-dialysis serumphosphorus concentration, depending on both dialyzer phosphate clearanceand the patient-specific K_(M). Further, short daily hemodialysis withreduced dialyzer phosphate clearance does not result in reductions inpre-dialysis serum phosphorus concentration unless treatment time isincreased substantially. The inventors conclude that increasing bothdialyzer phosphate clearance and treatment time during short dailyhemodialysis can result in clinically significant reductions inpre-dialysis serum phosphorus concentration.

An additional example of the use of this steady state mass balance modelis its application to determining optimal hemodialysis prescriptionsduring frequent nocturnal hemodialysis (e.g. 6-times per week, 8 hoursper treatment). During this therapy, K_(D)s are often empiricallylowered by adding phosphate salt supplements to the dialysis solution inorder to maintain pre-dialysis serum phosphorus concentrations within anoptimal range; however, there are no quantitative guides to determine anoptimal K_(D). The inventors used the above model to determine K_(D)that maintains pre-dialysis serum phosphorus concentration within therange recommended by the Dialysis Outcomes and Practice Patters Study(“DOPPS”) of 3.6-5.0 mg/dl. Computer simulations were performed for agiven dietary phosphorus intake (e.g., assuming no use of oral binders),post-dialytic phosphorus distribution volume of 12 L, and net fluidremoval per treatment of 1 L.

Calculated ranges for K_(D) (ml/min) to maintain pre-dialysis serumphosphorus concentration between 3.6 and 5.0 mg/dl for hypotheticalpatients with different K_(M) at steady state are tabulated in TableII.6.

TABLE II.6 K_(M) (ml/min) Dietary P Intake 50 100 150 4 g/week 40-70 35-53 32-48 5 g/week 57-114 46-75 42-65 6 g/week 81-200  60-102 54-84

These simulations demonstrate that individualization of K_(D), dependingon both dietary intake of phosphorus and the patient-specific K_(M), isrequired during frequent nocturnal hemodialysis.

III. Potassium Modeling

Potassium Prediction Methods in Hemodialysis Patients and ApplicationsThereof

In light of the systems discussed herein, it is also contemplated toprovide methods of predicting serum or plasma potassium concentrationsor levels in a hemodialysis patient before, during and afterhemodialysis therapies. Being able to predict serum potassium levels canbe useful in determining optimal treatment regimens for hemodialysispatients. These methods can be incorporated into any of the systems andcomputing devices described herein to optimize hemodialysis therapiesfor the patient.

It is important to control serum potassium levels within a certainrange, especially in chronic kidney disease stage 5 patients, as bothhyperkalemia and hypokalemia are associated with a greater risk ofmortality. Hyperkalemia refers to the condition in which serum potassiumlevels are too high, and hypokalemia refers to the condition in whichserum potassium levels are too low. The optimal predialysis serumpotassium concentration in chronic kidney disease stage 5 patients hasbeen suggested to be 4.6 to 5.3 mEq/L, as compared to a normal range of3.5 to 5.0 mEq/L.

Methods of predicting or determining serum potassium levels of a patientundergoing hemodialysis using a robust or practical potassium kineticmodel allow for effectively modifying new HD treatment modalities on anindividual patient basis. The methods of predicting or determining serumpotassium levels of a patient undergoing hemodialysis are similar to themethods of predicting or determining serum phosphate levels discussedabove and use a robust and practical potassium kinetic model to allowfor effectively modifying new HD treatment modalities on an individualpatient basis. In an embodiment, a method of predicting serum potassiumconcentration in a patient during hemodialysis is provided. The methodincludes measuring serum potassium concentrations (“C”) of the patientover a hemodialysis treatment session time and an ultrafiltration orfluid removal rate (“Q_(UF)”) calculated by a difference between pre-and post-dialytic body weight of the patient during an initialhemodialysis treatment session divided by the total treatment time ofthe treatment session and estimating K_(M) and V_(PRE) for the patientusing a non-linear least squares fitting to the governing transportequations having analytical solutions of the following form:

$\begin{matrix}{{{C(t)} = {C_{PRE}\begin{bmatrix}{\frac{K_{M} + {D\; {C_{D}/C_{PRE}}}}{K_{M} + K_{R} + D - Q_{UF}} +} \\{\left\lbrack {1 - \frac{K_{M} + {D\; {C_{D}/C_{PRE}}}}{K_{M} + K_{R} + D - Q_{UF}}} \right\rbrack \times} \\\left\lbrack \frac{V(t)}{V_{PRE}} \right\rbrack^{\frac{K_{M} + K_{R} + D - Q_{UF}}{Q_{UF}}}\end{bmatrix}}}{and}} & \left( {{III} - A} \right) \\{{C(T)} = {C_{PRE} - {\left( {C_{PRE} - C_{POST}} \right)e^{({- \frac{K_{M}T}{V_{PRE} - {Q_{UF}t_{tx}}}})}}}} & \left( {{III} - B} \right)\end{matrix}$

wherein t is a time during the hemodialysis treatment session, T is atime after an end of the hemodialysis treatment session, t_(tx) is atotal duration of the hemodialysis treatment session, C_(PRE) is apre-dialysis plasma potassium concentration, C_(POST) is a post-dialyticplasma potassium concentration, K_(M) is a potassium mobilizationclearance of the patient, K_(R) is a residual renal clearance ofpotassium, C_(D) is the dialysate potassium concentration, D is thedialyzer potassium dialysance, V_(PRE) is a pre-dialysis distributionvolume of potassium of the patient, and

V(t)=V _(PRE) −Q _(UF) ×t  (III-C).

C (i.e., serum potassium concentrations) of the patient can then bepredicted at any time during any hemodialysis treatment session by usingequations III-A and III-B, for the previously estimated set of K_(M) andV_(PRE) of the patient. t_(tx) can be any suitable amount of time suchas, for example, 2, 4 or 8 hours. T can be any suitable time, forexample, such as 30 minutes or 1 hour. Alternative to the non-linearleast squares fitting, V_(PRE) can also be estimated as a certainpercentage of body weight or body water volume of the patient.

Like with phosphorus, a method of predicting serum potassiumconcentration in a patient during hemodialysis can also be provided whenthe ultrafiltration rate is assumed to be negligible (i.e., Q_(UF)=0).The method includes measuring C of the patient during an initialhemodialysis treatment session and estimating K_(M) and V_(PRE) for thepatient using a non-linear least squares fitting to the governingtransport equations having analytical solutions of the following form:

$\begin{matrix}{{{C(t)} = {C_{PRE}\begin{bmatrix}{\frac{K_{M} + {D\; {C_{D}/C_{PRE}}}}{K_{M} + K_{R} + D} +} \\{\left\lbrack {1 - \frac{K_{M} + {D\; {C_{D}/C_{PRE}}}}{K_{M} + K_{R} + D}} \right\rbrack e^{({- \frac{K_{M} + K_{R} + D}{V_{PRE}}})}}\end{bmatrix}}}{and}} & \left( {{III} - D} \right) \\{{C(T)} = {C_{PRE} - {\left( {C_{PRE} - C_{POST}} \right){e^{({- \frac{K_{M}T}{V_{PRE}}})}.}}}} & \left( {{III} - E} \right)\end{matrix}$

C of the patient can be predicted at any time during any hemodialysistreatment session, for example every 15 or 30 minutes, by usingequations III-D and III-E for a given set of previously estimatedparameters, K_(M) and V_(PRE), of the patient. T can be any suitabletime, for example, such as 30 minutes or 1 hour. Alternatively, V_(PRE)can be further estimated as a certain percentage of body weight or bodywater volume of the patient. In an embodiment, K_(M) may be estimatedusing data from a case where Q_(UF)≠0 and used in equation III-D whereQ_(UF)=0.

The methods of predicting or determining serum potassium levels thendiffers from the methods of predicting or determining serum phosphatelevels described above in that the calculation of dialyzer potassiumdialysance (D) is determined using the equation:

$\begin{matrix}{{D = {Q_{B}\frac{\left( {0.94 - {{Hct} \times 100}} \right)\left( {e^{Z} - 1} \right)}{\left( {e^{Z} - \frac{\left( {0.94 - {{Hct} \times 100}} \right)Q_{B}}{Q_{D}}} \right)}}}{wherein}} & \left( {{III} - F} \right) \\{{Z = {K_{O}A\frac{\left( {Q_{D} - {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B}}} \right)}{\left( {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B} \times Q_{D}} \right)}}},} & \left( {{III} - G} \right)\end{matrix}$

wherein Q_(B) and Q_(D) are the blood and dialysate flow rates at whichthe desired dialyzer potassium dialysance D is calculated usingequations III-F and III-G. In calculating D, it can be assumed in oneembodiment that the dialyzer mass transfer-area coefficient (K₀A) forpotassium is 80% or about 80% of that for urea. Alternatively, K₀A canbe calculated using the following equation:

$\begin{matrix}{{K_{O}A} = {\frac{\left( {0.94 - {{Hct} \times 100}} \right)Q_{B,M} \times Q_{D,M}}{Q_{D,M} - {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B,M}}} \times {{\ln \left( \frac{1 - {D_{M}/Q_{D,M}}}{1 - {D_{M}/\left\lbrack {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B,M}} \right\rbrack}} \right)}.}}} & \left( {{III} - H} \right)\end{matrix}$

In equation III-H, K_(O)A is a dialyzer mass transfer area coefficientfor potassium obtained as a result of a previous measurement where theset of blood and dialysate flow rates Q_(B,M) and Q_(D,M) resulted in ameasured or assumed dialyzer dialysance D_(M), and Hct is hematocritcount measured from a patient's blood sample. Otherwise, the estimationof the kinetic parameters for potassium proceeds as described previouslyfor phosphorus.

Alternatively, equations II-A through II-E can be used if a dialyzerclearance K_(D) is determined using the following equation, whichaccounts for non-zero dialysate potassium concentrations:

$\begin{matrix}{K_{D} = {\frac{D}{2}\left( {2 - \frac{C_{D}}{C_{PRE}} - \frac{C_{D}}{C_{POST}}} \right)}} & \left( {{III} - I} \right)\end{matrix}$

wherein C_(PRE) is the pre-dialysis serum potassium concentration,C_(POST) is the post-dialysis serum potassium concentration, and C_(D)is the dialysate potassium concentration.

Alternative to non-linear least squares fitting, K_(M) can be determinedusing the following algebraic equation:

$\begin{matrix}{K_{M} = {{C_{POST}\left( \frac{K_{D} - Q_{UF}}{C_{PRE} - C_{POST}} \right)}.}} & \left( {{III} - J} \right)\end{matrix}$

V_(POST) is a measure of the distribution volume of potassium at the endof the hemodialysis treatment when the patient is considered to benormohydrated. This parameter approximates the volume of extracellularfluids. Thus, V_(POST) is a clinically relevant patient parameter thatcan be used to evaluate the patient's hydration status. In anapplication from knowing the previously determined V_(PRE), V_(POST) canbe determined using the equation:

V _(POST) =V _(PRE) −Q _(UF) ×t _(tx)  (III-K)

and a suitable therapy can be provided to the patient based on the valueof V_(POST). As seen from equation III-K, if Q_(UF)=0, thenV_(POST)=V_(PRE).

Specific steps of the methods of predicting potassium mobilization in apatient during hemodialysis can be performed using a computing device.Such a computing device can include a display device, an input device, aprocessor, and a memory device that stores a plurality of instructions,which when executed by the processor, cause the processor to operatewith the display device and the input device to (a) receive datarelating to C of a hemodialysis patient over a hemodialysis treatmentsession time and a Q_(UF) calculated based on a difference between pre-and post-dialytic body weight of the hemodialysis patient during ahemodialysis treatment session divided by the total treatment time ofthe treatment session; (b) estimate K_(M) and V_(PRE) for the patientusing a non-linear least squares fitting to the governing transportequations having analytical solutions of the form of equations III-A andIII-B (or alternative equivalents); and (c) predict C of the patient atany time during hemodialysis by using the equations III-A and III-B fora given set of estimated parameters, K_(M) and V_(PRE), of the patient.It should be appreciated that the variables for equations III-A andIII-B can be determined using any of the equations set forth herein. Theinformation/data obtained for the hemodialysis patient can bedisplayed/printed out and used by the healthcare provider to provideimproved treatment and nutritional regimens for the hemodialysispatient. Any of the unknown factors can be determined using theappropriate equations or measurements discussed previously for themethods of determining potassium mobilization in a patient duringhemodialysis. If Q_(UF)=0, then equations III-D and III-E (oralternative equivalents) should be used.

The computing device can also be preprogrammed or run according tosoftware that causes the processor to operate with the display deviceand the input device to receive data relating to at least one of K_(R),K_(D) or a sampling time for collecting the serum potassiumconcentration. In an embodiment, the computing device can be system 10described in section I.

Along with the previously described methods of determining potassiummobilization in a patient during hemodialysis, a mass balance model topredict steady state, pre-dialysis serum potassium levels (“C_(SS-PRE)”)in patients treated with HD therapies has also been developed. A massbalance model can be used in combination with a pseudo one-compartmentmodel for intradialytic and rebound periods to determine steady statepre-dialysis serum potassium levels in individual patients. Using thismodel, the effect of specific therapy parameters (e.g., dialyzerpotassium clearances, weekly therapy frequency, therapy duration, etc.)on individual hemodialysis patients' serum potassium levels can beevaluated. The kinetics of potassium during HD therapies cannot bedescribed using a conventional one-compartment model because theinterdialytic decreases in serum potassium concentration are differentthan those for urea, and there is a substantial post-dialysis rebound ofpotassium concentration. The kinetics of potassium during HD therapieshave been previously described using a two-compartment model assumingthat the distribution of potassium is confined to classic intracellularand extracellular fluid compartments with both active and passivetransport of potassium between the compartments, but suchtwo-compartment models are complex and require a relatively large numberof parameters to describe potassium kinetics.

The disclosed steady state, mass balance model combines theintradialytic potassium kinetics with dietary intake, use of potassiumbinders, and residual renal clearance to predict steady state,pre-dialysis serum potassium levels. Unlike those with previous models,the predictions with this model involve simplified calculations; hence,this model can easily be integrated in daily clinical practice.Furthermore, the model involves patient-specific parameters enablingindividualized predictions. This model can eventually be used tooptimize therapies with a HHD device to remove adequate amounts ofpotassium using minimum necessary volumes of dialysate (i.e., minimizedwater consumption). Alternatively, the model can be used to determinethe amount of required potassium to add to the dialysate.

In an application of the kinetic model, a method of determining theC_(SS-PRE) in a hemodialysis patient is provided. The method includesobtaining a net generation of potassium (“G”) from at least a dietarypotassium intake of the patient and determining C_(SS-PRE) of thehemodialysis patient using the equation:

$\begin{matrix}{C_{{SS} - {PRE}} = \frac{{G\left( {10080/F} \right)} + {D \times C_{D} \times t_{tx}}}{{\left( {D + K_{R}} \right)n{\overset{\_}{C}}_{tx}t_{tx}} + {K_{R}n{{\overset{\_}{C}}_{i}\left( {{10080/F} - t_{tx}} \right)}}}} & \left( {{III} - L} \right)\end{matrix}$

wherein C_(D) is the dialysate potassium concentration, K_(R) is aresidual renal clearance of potassium, t_(tx) is a total duration of thehemodialysis treatment session, D is the dialyzer dialysance, F is afrequency of treatments per week, n C _(tx) is the normalized timeaveraged plasma potassium concentration during a dialysis treatment, andn C _(i) is the normalized time averaged plasma potassium concentrationfor an interdialytic interval. This equation takes into accountpotassium in the dialysate. The effect of at least one of a patientparameter or a treatment parameter on C_(SS-PRE) of the patient can besimulated so as to obtain an optimal range of C_(SS-PRE) for thepatient.

In an alternative embodiment, a method of predicting the C_(SS-PRE) in ahemodialysis patient is provided. The method includes determining a netgeneration of potassium (“G”) using the equation:

$\begin{matrix}{G = {\left\lbrack \frac{{\left( {D + K_{R}} \right)n{\overset{\_}{C}}_{tx}t_{tx}} + {K_{R}n{{\overset{\_}{C}}_{i}\left( {{10080/F} - t_{tx}} \right)}}}{\left( {10080/F} \right)} \right\rbrack \times {\quad\begin{bmatrix}{C_{{SS} - {PRE} - {IN}} -} \\\frac{D \times C_{D} \times t_{tx}}{{\left( {D + K_{R}} \right)n{\overset{\_}{C}}_{tx}t_{tx}} + {K_{R}n{{\overset{\_}{C}}_{i}\left( {{10080/F} - t_{tx}} \right)}}}\end{bmatrix}}}} & \left( {{III} - M} \right)\end{matrix}$

wherein C_(SS-PRE-IN) is an initial, measured, steady state,pre-dialysis serum potassium level of the hemodialysis patient who ismaintained by a hemodialysis therapy for a specified time prior to thecalculation of G using equation III-M. The specified time can be, forexample, at least one week, two weeks, three weeks, 1 month, 2 months, 3months, 4 months or more prior to the time when G is calculated.

Once G has been calculated using equation III-M or estimated by othermethods, it can be used in predicting the effect of changes inhemodialysis treatment parameters on the steady state serum potassiumconcentration. For example, once G of the hemodialysis patient is known,C_(SS-PRE) of the patient under different hemodialysis treatmentconditions can be predicted by rearranging equation III-M to formequation III-L and utilizing the known G to solve for C_(SS-PRE) of thehemodialysis patient. The effect of at least one of a patient parameteror a treatment parameter on C_(SS-PRE) of the patient can be simulated,and a treatment regimen of the hemodialysis patient can then be modifiedso that C_(SS-PRE) is within a desired range.

In general, there is an optimal range of steady state, pre-dialysisserum potassium levels in patients with end stage renal disease. Optimalprescription/regimen/nutritional therapies resulting in steady state,pre-dialysis potassium levels within the desired optimal ranges can bedetermined using equations III-L and III-M, for example, in theOptimization Component of the HHD system previously described herein.Because changes in the hemodialysis prescription or in patient behavior(e.g., changes in diet) could lead to changes in G, optimization of homehemodialysis therapies based on equations III-L and III-M to maintainC_(SS-PRE) within a desired range is advantageous.

In any of the methods of determining the G or the C_(SS-PRE) in ahemodialysis patient, n C _(tx) and n C _(i) can be determined using theequations:

$\begin{matrix}{{{n{\overset{\_}{C}}_{tx}} = {\frac{1}{t_{tx}}\begin{Bmatrix}{\left\lbrack \frac{\left( {K_{M} + {D\; {C_{D}/C_{PRE}}}} \right)t_{tx}}{K_{M} + K_{R} + D - Q_{UF}} \right\rbrack +} \\{{\left\lbrack {1 - \frac{K_{M} + {D\; {C_{D}/C_{PRE}}}}{K_{M} + K_{R} + D - Q_{UF}}} \right\rbrack \left\lbrack \frac{V_{PRE}}{K_{M} + K_{R} + D} \right\rbrack}\left\lbrack {1 - \left( \frac{V_{POST}}{V_{PRE}} \right)^{\frac{({K_{M} + K_{R} + D})}{Q_{UF}}}} \right\rbrack}\end{Bmatrix}}},{and}} & \left( {{III} - N} \right) \\{{{n{\overset{\_}{C}}_{i}} = {\frac{1}{{10080/F} - t_{tx}}\begin{Bmatrix}{\left\lbrack \frac{K_{M}\left( {{10080/F} - t_{tx}} \right)}{K_{M} + K_{R} + Q_{WG}} \right\rbrack +} \\{\begin{bmatrix}{\frac{K_{M} + {D\; {C_{D}/C_{PRE}}}}{K_{M} + K_{R} + D - Q_{UF}} +} \\{{\left\lbrack {1 - \frac{K_{M} + {D\; {C_{D}/C_{PRE}}}}{K_{M} + K_{R} + D - Q_{UF}}} \right\rbrack \left\lbrack \frac{V_{POST}}{V_{PRE}} \right\rbrack}^{\frac{({K_{M} + D + K_{R} - Q_{UF}})}{Q_{UF}}} -} \\\frac{K_{M}}{K_{M} + K_{R} + Q_{WG}}\end{bmatrix} \times} \\{\left\lbrack \frac{V_{POST}}{K_{M} + K_{R}} \right\rbrack \left\lbrack {1 - \left( \frac{V_{POST}}{V_{PRE}} \right)^{\frac{({K_{M} + K_{R}})}{Q_{WG}}}} \right\rbrack}\end{Bmatrix}}},} & \left( {{III} - O} \right)\end{matrix}$

wherein Q_(WG) is a constant rate of fluid gain by the patient duringthe interdialytic time interval. In any of the methods of determining Gor C_(SS-PRE) in a hemodialysis patient when there is negligible netultrafiltration or fluid removal from the patient during hemodialysistherapies and no weight gain between hemodialysis therapies, n C _(tx)and n C _(i) can be determined using the equations:

$\begin{matrix}{{{n{\overset{\_}{C}}_{tx}} = {\frac{1}{t_{tx}}\begin{Bmatrix}{\left\lbrack \frac{\left( {K_{M} + {D\; {C_{D}/C_{PRE}}}} \right)t_{tx}}{K_{M} + K_{R} + D} \right\rbrack +} \\{{\left\lbrack {1 - \frac{K_{M} + {D\; {C_{D}/C_{PRE}}}}{K_{M} + K_{R} + D}} \right\rbrack \left\lbrack \frac{V_{PRE}}{K_{M} + K_{R} + D} \right\rbrack}\left\lbrack {1 - e^{(\frac{{- {({D + K_{R} + K_{M}})}}t_{tx}}{V_{PRE}})}} \right\rbrack}\end{Bmatrix}}},{and}} & \left( {{III} - P} \right) \\{{n{\overset{\_}{C}}_{i}} = {\frac{1}{{10080/F} - t_{tx}}{\begin{Bmatrix}{\left\lbrack \frac{K_{M}\left( {{10080/F} - t_{tx}} \right)}{K_{M} + K_{R}} \right\rbrack +} \\{\begin{bmatrix}{\frac{K_{M} + {D\; {C_{D}/C_{PRE}}}}{K_{M} + K_{R} + D} +} \\\begin{matrix}{\left\lbrack {1 - \frac{K_{M} + {D\; {C_{D}/C_{PRE}}}}{K_{M} + K_{R} + D}} \right\rbrack \times} \\{e^{\frac{{- {({D + K_{R} + K_{M}})}}t_{tx}}{V_{PRE}}} -}\end{matrix} \\\frac{K_{M}}{K_{M} + K_{R}}\end{bmatrix} \times} \\{\left\lbrack \frac{V_{POST}}{K_{M} + K_{R}} \right\rbrack \left\lbrack {1 - e^{(\frac{{- {({K_{M} + K_{R}})}}{({{10080/F} - t_{tx}})}}{V_{POST}})}} \right\rbrack}\end{Bmatrix}.}}} & \left( {{III} - Q} \right)\end{matrix}$

It should be appreciated that the variables for equations III-P andIII-Q can be determined using any of the equations set forth herein.

In an embodiment, K_(M) and V_(PRE) can be determined using the methodsof predicting potassium mobilization in a patient during hemodialysis aspreviously discussed. In this case, K_(M) and V_(PRE) are determined bymeasuring C of the hemodialysis patient over a hemodialysis treatmentsession time and Q_(UF) calculated by a difference between pre- andpost-dialytic body weight of the patient during an initial hemodialysistreatment session divided by the total treatment time of the treatmentsession, and by estimating K_(M) and V_(PRE) for the hemodialysispatient using a non-linear least squares fitting to the governingtransport equations having analytical solutions that follow fromequations III-A and III-B.

The methods of determining G or C_(SS-PRE) in a hemodialysis patient canalso be used to determine or modify the appropriate treatments/dietarychanges to meet a desired potassium serum level in the hemodialysispatient over a period of time. For example, the methods can be used todetermine or modify a level of potassium intake so that C_(SS-PRE) ofthe hemodialysis patient ranges between about 3.5 and 5.0 mEq/L (normalrange) or about 4.6 and 5.3 mEq/L (suggested optimal range). The methodscan be used to determine or modify a potassium binder administered tothe patient so that C_(SS-PRE) of the hemodialysis patient rangesbetween about 3.5 and 5.0 mEq/L or about 4.6 and 5.3 mEq/L. The methodscan further be used to determine or modify an amount of potassium addedto the dialysate so that C_(SS-PRE) of the hemodialysis patient rangesbetween about 3.5 and 5.0 mEq/L or about 4.6 and 5.3 mEq/L.

The methods can be used to determine or modify the total hemodialysistreatment session time so that C_(SS-PRE) of the hemodialysis patientranges between about 3.5 and 5.0 mEq/L or about 4.6 and 5.3 mEq/L. Themethods can be used to determine or modify the frequency F so thatC_(SS-PRE) of the hemodialysis patient ranges between about 3.5 and 5.0mEq/L or about 4.6 and 5.3 mEq/L. The methods can be used to determineor modify a required blood flowrate and/or a dialysate flowrate so thatC_(SS-PRE) of the hemodialysis patient ranges between about 3.5 and 5.0mEq/L or about 4.6 and 5.3 mEq/L. It should be appreciated that thepreferred range of C_(SS-PRE) can be patient specific.

Specific steps of determining C_(SS-PRE) of a hemodialysis patient canbe performed using a computing device. Such a computing device caninclude a display device, an input device, a processor, and a memorydevice that stores a plurality of instructions, which when executed bythe processor, cause the processor to operate with the display deviceand the input device to: (a) receive data relating to G from, forexample, a dietary potassium intake of a hemodialysis patient; (b)determine the C_(SS-PRE) of the patient using equation III-L; and (c)simulate the effect of at least one of a patient parameter or atreatment parameter on C_(SS-PRE) of the hemodialysis patient. It shouldbe appreciated that the variables for equation III-L can be determinedusing any of the appropriate equations set forth herein.

Another such computing device can include a display device, an inputdevice, a processor, and a memory device that stores a plurality ofinstructions, which when executed by the processor, cause the processorto operate with the display device and the input device to: (a)determine a net generation of potassium (“G”) using equation III-M; (b)predict steady state, pre-dialysis serum potassium levels (“C_(SS-PRE)”)of the hemodialysis patient using equation III-L; and (c) simulate theeffect of at least one of a patient parameter or a treatment parameteron C_(SS-PRE) of the hemodialysis patient. It should be appreciated thatthe variables for equations III-L and III-M can be determined using anyof the appropriate equations or methods set forth herein.

In any of the computing devices described herein, the information/dataobtained for the hemodialysis patient can be displayed/printed out andused by the healthcare provider to provide improved treatment andnutritional regimens for the hemodialysis patient. Any of the unknownfactors can be determined using any of the appropriate equations ormeasurements discussed herein for the methods of determining the steadystate, pre-dialysis serum potassium levels of a hemodialysis patient.

The computing devices can also be preprogrammed or run according tosoftware that causes the processor to operate with the display deviceand the input device to receive data relating to at least one of K_(R),K_(D), D, K_(M), V_(PRE), t_(tx), F, and C_(PRE) about a month before ahemodialysis treatment session or a sampling time for collecting theserum potassium concentration. The computer device utilizes thisinformation to simulate the effect of one or more of these patientparameters or treatment parameters on C_(SS-PRE) of the hemodialysispatient, for example, using equation III-L (e.g., seeing how a change inone or more of the patient parameters or treatment parameters impactsC_(SS-PRE)). The computing device can be preprogrammed to display atreatment regimen of the hemodialysis patient so that C_(SS-PRE) iswithin a desired range using any of the methods disclosed herein. In anembodiment, the computing device can be system 10 described in sectionI.

Any of the computer devices described herein (including any portions ofsystem 10 described in section I) can be a device having a processorcapable of receiving data and performing calculations based on thatdata. Such computing device can be, for example, a handheld clientdevice, personal computer client device, database server, etc.). A moredetailed block diagram of the electrical systems of the computingdevices described herein is illustrated in FIG. 10.

EXAMPLES

By way of example and not limitation, the following examples areillustrative of various embodiments of the present disclosure andfurther illustrate experimental testing conducted with the systems andmethods in accordance with embodiments of the present disclosure.

Example 1 Objective

The objective of this analysis was to demonstrate the procedure forestimating specific parameters (e.g., K_(M) and V_(PRE)) from a pseudoone-compartment model using clinical data, and to evaluate the validityof parameter estimates over different HD treatment modalities. Thekinetics of potassium during hemodialysis cannot be described using aconventional one-compartment model because the interdialytic decreasesin serum potassium concentration are different than those for urea, andthere is a substantial post-dialysis rebound of potassium concentration.The pseudo one-compartment model is also advantageous overtwo-compartment models, which allow for two-compartment (intracellularand extracellular) volumes of distribution and active transport ofpotassium, but are complex and require a relatively large number ofparameters to describe potassium kinetics.

The pseudo one-compartment model can be used to evaluate theintradialytic and postdialytic rebound potassium kinetics and includesonly two kinetic parameters (K_(M) and V_(PRE)). This model isadvantageous over previous models that assume potassium distribution inconventional intracellular and extracellular compartments, incorporateactive sodium-potassium pump activity, and contain numerous kineticparameters that cannot be uniquely estimated from limited data during HDtreatments when only measuring the dependence of serum potassiumconcentration on time.

Psuedo One-Compartment Model

Potassium is not uniformly distributed in body fluids. Less than 2% oftotal body potassium is contained with the extracellular compartment,and the remaining 98% is intracellular. Such a distribution of bodypotassium suggests that its kinetics during hemodialysis can bedescribed using a pseudo one-compartment kinetic model, in whichpotassium is removed during hemodialysis from a central compartment (theextracellular space) that is much smaller than a second inaccessiblecompartment, analogous to that previously demonstrated for phosphorus.

A conceptual description of a pseudo one-compartment kinetic model forpotassium is shown in FIG. 25. In this model, potassium is removed bythe dialyzer from a compartment of volume V (also called thedistribution volume) and potassium concentration C. The distributionvolume is assumed to be in equilibrium with plasma. Potassiummobilization into this compartment occurs from potassium compartments inthe body that are inaccessible to the dialyzer. The dashed linesindicate that the peripheral or inaccessible compartment volume has beenassumed to be very large such that the potassium concentration in thecompartment remains constant at its predialysis value (“C_(PRE)”). Thepotassium mobilization clearance is denoted by K_(M) and the dialyzerpotassium clearance is denoted by K_(D).

Changes in the volume and potassium concentration of a potassiumdistribution volume during and shortly after an HD treatment session arerepresented by equations E-D1 and E-D2,

$\begin{matrix}{\frac{({VC})}{t} = {{K_{M}\left( {C_{PRE} - C} \right)} - {K_{R}C} - {D\left( {C - C_{D}} \right)}}} & {E - {D1}} \\{\frac{(V)}{t} = {{- \Theta}\; Q_{UF}}} & {E - {D2}}\end{matrix}$

wherein C is the serum or plasma concentration of potassium, C_(PRE) isthe predialysis serum concentration of potassium, C_(D) is the dialysatepotassium concentration, D is the dialyzer potassium dialysance, K_(M)is a potassium mobilization clearance of the patient, and K_(R) is theresidual renal or kidney clearance. Equation E-D1 differs from the massbalance model for phosphorus by substituting dialysance for clearanceand allowing for a non-zero value for C_(D). Θ is a variable thatindicates whether dialysis treatment is taking place (Θ=1) or not (Θ=0),and Q_(UF) is the ultrafiltration (“UF”) rate. The kinetic modeldescribed above also assumes that all fluid removed during the treatmentis from the distribution volume of potassium.

Closed form analytical solutions to the time-dependent plasma potassiumconcentration can be obtained by integrating equations 1 and 2. For theintradialytic (Θ=1) and rebound (Θ=0) periods, the time dependence ofpotassium concentration can be expressed as shown by equations E-D3 andE-D4 respectively:

$\begin{matrix}{{C(t)} = {C_{PRE}\left\lbrack {\frac{K_{M} + {{DC}_{D}/C_{PRE}}}{K_{M} + K_{R} + D - Q_{UF}} + \left. \quad{\left\lbrack {1 - \frac{K_{M} + {{DC}_{D}/C_{PRE}}}{K_{M} + K_{R} + D - Q_{UF}}} \right\rbrack \times \left\lbrack \frac{V(t)}{V_{PRE}} \right\rbrack^{\frac{K_{M} + K_{R} + D - Q_{UF}}{Q_{UF}}}} \right\rbrack} \right.}} & {E\text{-}D\; 3} \\{\mspace{79mu} {{C(T)} = {C_{PRE} - {\left( {C_{PRE} - C_{POST}} \right)^{({- \frac{K_{M}T}{V_{PRE} - {Q_{UF}t_{tx}}}})}}}}} & {E\text{-}D\; 4}\end{matrix}$

wherein V_(PRE) is the pre-dialytic distribution volume of potassium, tis the time during the treatment, T is the time after the end of thetreatment, and t_(tx) is total duration of treatment prior to therebound period. It is also assumed that distribution volume of potassiumremains constant during the post-dialytic rebound period.

Methods

Clinical data was obtained from a prospective, randomized, multicenterclinical trial to evaluate the effect of increased dialysis dose and theuse of high flux dialysis on patient outcomes. A total of 1846 patientswere randomized between 1995 and 2001 in 72 dialysis units affiliatedwith 15 clinical centers in the United States.

During the execution of this clinical trial, kinetic modeling sessionswere performed during months 4 and 36 of patient follow-up to betterunderstand and monitor adherence to the dose intervention. Midwaythrough the trial, other chemical assays in addition to urea wereperformed on the collected serum samples to explore the kinetics ofsolutes other than urea, one of which was potassium. Kinetic modelingsessions were performed using four different blood samples: (1) a bloodsample taken during predialysis; (2) a blood sample take 60 minutesafter starting the treatment; (3) a blood sample taken 20 seconds afterstopping the treatment using a slow flow technique; and (4) a bloodsample taken 30 minutes after the end of treatment. All samples wereallowed to clot and then centrifuged to obtain serum for chemicalanalysis. Serum samples were sent to a central laboratory for assay ofpotassium concentration (recorded to the nearest 0.1 mEq/L).

During months 4 and 36 of follow-up 792 patients had serum potassiumconcentrations measured on all 4 serum samples. Data was excluded if apatient had any serum potassium concentration greater than 10 mEq/L orequal to 0 mEq/L, or had a unexpectedly high postdialytic serumpotassium concentration (postdialytic-to-predialytic concentration ratio≧1.2).

During each kinetic modeling session, dialyzer potassium clearance wasestimated using the recorded blood and dialysate flow rates, the plasmaand dialysate potassium concentrations, and the effective diffusion flowrates assuming the dialyzer mass transfer area coefficient for potassiumwas 80% of that determined for urea at the specified dialysate flow rateand that the hematocrit was 33% for all patients. The ultrafiltrationrate during the treatment was assumed constant and calculated as thepredialysis body weight minus the postdialysis body weight divided bythe treatment time. It was assumed that all fluid removed during thetreatment was from the central distribution volume of potassium.

Kinetic analysis was performed using the model shown in FIG. 25 bydetermining the best fit between the predicted potassium concentrationsat each time with those measured experimentally as described previouslyfor analyzing phosphorus kinetics. Optimal values of potassiummobilization clearance (K_(M)) and predialysis central distributionvolume of potassium (V_(PRE)) and their standard errors weresimultaneously estimated for each patient using nonlinear regression.Assuming comparable errors in each concentration measurement, thepredialysis serum potassium concentration was also treated as an unknownparameter. K_(M) or V_(PRE) estimates with standard errors 5 timesgreater than the optimal values were further excluded from the data set,as were those with K_(M) values less than zero and V_(PRE) values lessthan 0.1 L, leaving a total of 551 patients for further analysis.Because estimated values of predialysis potassium concentration are notphysiologically of interest, they have not been described further.

Potassium removal was calculated using the trapezoid method as the areaunder the serum concentration curve times the dialyzer potassiumconcentration. All values are reported as mean±standard deviation ormedian (interquartile range). Simple and stepwise multiple linearregression of the logarithm of K_(M) and V_(PRE) with patientcharacteristics and other measured variables were performed usingcommercial software.

For all patients, the serum potassium concentration decreased frompredialysis at 5.3±0.9 mEq/L to 4.1±0.7 mEq/L after 60 minutes oftreatment and to 3.6±0.6 mEq/L postdialysis. Serum potassiumconcentration rebounded to 4.1±0.7 mEq/L at 30 minutes postdialysis.K_(M) and V_(PRE) for all patients had median values of 158 ml/min and15.6 L, respectively. Mean absolute differences between the measured andmodel-predicted serum potassium concentrations for all patients were0.01 mEq/L. K_(M) and Vpp were not normally distributed, and thereforelogarithmic transformation was applied to both kinetic parameters priorto statistical testing or modeling. FIG. 26 shows the distribution ofthese estimated kinetic parameters for all patients. More specifically,FIG. 26 shows the predialysis potassium central distribution volume(V_(PRE)) versus the potassium mobilization clearance (K_(M)) for allpatients on logarithm scales.

To examine the influence of dialysate potassium concentration on thekinetic parameters, the dialysate potassium concentration was dividedinto 4 nominal categories: 0K, 1K, 2K and 3K. Table III.1 compares thepatient and treatment characteristics and FIG. 27 shows the dependenceof serum potassium concentration versus time for the four nominaldialysate potassium concentrations. In Table III.1, 0K includes 0.0-0.5mEq/L, 1K includes 0.6-1.5 mEq/L, 2K includes 1.6-2.5 mEq/L, and 3Kincludes ≧2.6 mEq/L of potassium concentration.

TABLE III.1 Dialysate Potassium Concentration (mEq/L) Characteristic 0 K1 K 2 K 150 N 4 60 437 50 Age (years)  51.6 ± 16.5 54.5 ± 15.6 59.3 ±14.2 59.7 ± 14.5 Female Sex 2, 50 38, 63 242, 55 23, 46 (N, %)Predialytic 70.1 ± 3.7 71.6 ± 15.5 72.6 ± 15.0 68.4 ± 15.2 Body Weight(kg) Postdialytic 66.6 ± 5.1 68.5 ± 15.2 69.7 ± 14.6 65.4 ± 15.1 BodyWeight (kg) Treatment 200 ± 32 206 ± 27  206 ± 29  208 ± 30  Time (min)Blood Flow 361 ± 51 348 ± 87  370 ± 81  380 ± 85  Rate (mL/min)Dialysate  575 ± 150 642 ± 139 693 ± 127 700 ± 128 Flow Rate (mL/min)Dialyzer 179 ± 16 172 ± 29  182 ± 26  186 ± 30  Potassium Dialysance(mL/min) Dialyzer 179 ± 16 129 ± 21  95 ± 17 57 ± 17 Potassium Clearance(mL/min)

In general and on average, patients treated with low dialysate potassiumconcentrations had high predialysis serum concentrations and highintradialytic decreases in serum potassium concentration, as shown inFIG. 27. FIG. 27 also shows that the decrease in serum potassiumconcentration during the treatment was greater when the dialysatepotassium concentration was low.

Table III.2 summarizes estimates of K_(M) and V_(PRE) for the fournominal dialysate potassium concentration categories discussed above.Neither the potassium mobilization clearance nor the predialysis centraldistribution volume was a strong function of the dialysate potassiumconcentration. FIGS. 28 and 29 show the dependences of K_(M) and V_(PRE)on the initial dialysate potassium concentration. More specifically,FIG. 28 shows the mobilization clearance as a function of the initialdialysate potassium concentration, and FIG. 29 shows the predialysispotassium distribution volume as a function of the initial dialysatepotassium concentration. Values of K_(M) were independent of thedialysate potassium concentration, whereas V_(PRE) increased at a lowinitial dialysate potassium concentration. The data in these figuresdemonstrates the high overlap among the estimates of the kineticparameters over all dialysate potassium concentrations. The “Difference”shown in Table III.2 is the absolute difference between the measured andmodel-predicted concentrations.

TABLE III.2 Dialysate Potassium Median Value [Interquartile Range]Concentration Difference (mEq/L) K_(M) (mL/min) V_(PRE) (L) (mEq/L) 0 K143 [133-170] 21.1 [18.3-38.7] 0.11 [0.08-0.16] 1 K 177 [126-242] 20.0[13.2-29.2] 0.10 [0.04-0.14] 2 K 152 [111-242] 15.5 [11.7-21.9] 0.10[0.05-0.17] 3 K 161 [103-214] 12.1 [8.4-17.7]  0.07 [0.04-0.13]

The results demonstrate that the potassium mobilization clearance isindependent of the dialysate potassium concentration and that thecentral distribution volume is higher at low dialysate concentrations.As expected, potassium removal during the treatment was generally higherfor lower dialyzer potassium concentrations. Potassium removal for thedialysate potassium concentration categories were 123.5±9.9 mEq for 0K,69.5±25.8 mEq for 1K, 41.9±18.2 mEq for 2K, and 16.7±9.9 mEq for 3K.FIG. 30 plots the potassium removal divided by the predialysis serumconcentration separately for the four dialysate potassium concentrationcategories. There is significant overlap in this parameter for variousdialysate potassium concentrations, suggesting that potassium removalduring HD cannot be readily predicted by the predialysis serum potassiumconcentration.

The potassium mobilization clearance values determined in this study arecomparable to, but approximately 50% higher than, mobilizationclearances previously determined for phosphorus during hemodialysis.Potassium is a smaller electrolyte than phosphorus and therefore may bemore rapidly mobilized by differences in concentration induced by rapidremoval during hemodialysis. The potassium central distribution volumeis also comparable to, but larger than, that previously determined forphosphorus. The combined results suggest that both potassium andphosphorus have a kinetically-determined central distribution volumethat is comparable to, but greater than, the extracellular volume.Additional support for the applicability of the pseudo one-compartmentmodel is the finding that the potassium mobilization clearance isindependent of the dialysate potassium concentration. These resultssuggest that the kinetic parameters are relatively independent ofdialysate potassium concentration and are largely patient specific.

The present observations suggest that approximately half of thepotassium removed during hemodialysis comes from the intracellularcompartment, and further suggest that the amount of potassium removedfrom the inaccessible compartment will depend substantially on thepotassium mobilization clearance. The large variability in potassiummobilization clearance among patients (FIG. 27) and the effect of suchvariability on potassium removal (FIG. 30) demonstrate that thisparameter may be useful to quantify individual variability on serumpotassium concentrations and potassium removal among patients. Theresults suggest that K_(M) and V_(PRE) are patient specific parameters.The pseudo one-compartment model described above can therefore be usedto evaluate the intradialytic and postdialytic rebound potassiumkinetics using K_(M) and V_(PRE).

Example 2 Objective

A steady state mass balance model can also be based on the pseudoone-compartment model. The mathematical model can be used to further thecurrent understanding of potassium removal in hemodialysis patients andto predict optimal dialysate potassium concentrations for adequatedialytic removal of potassium while maintaining serum potassium withinthe normal range. Such models are especially advantageous during highdose hemodialysis therapies using frequent and long treatment sessions.

Similar to with phosphorus, a steady state potassium mass balance modelallows prediction of total potassium removal from the knowledge of: 1)the pre-dialysis concentration of potassium in plasma or serum, 2) thedialyzer clearance or dialysance of potassium, 3) the volume ofdistribution of potassium, 4) the amount of fluid removed during thetreatment, 5) dialysate concentration of potassium, and 6) apatient-specific potassium mobilization clearance. The steady statepotassium mass balance model described below will be used in combinationwith the previous kinetic model to allow determination of pre-dialysisserum potassium concentration for individual patients under anyhemodialysis treatment prescription when the above parameters 2-6 areestablished, the frequency of hemodialysis treatments per week and thehemodialysis treatment duration are prescribed, and the net generationof potassium (defined below), and residual kidney or renal potassiumclearance are all known. Alternatively, the steady state potassium massbalance model in combination with the previous kinetic model can be usedto determine the net generation of potassium for a given patient whenthe above parameters 1-6 are established and the frequency ofhemodialysis treatments per week, the hemodialysis treatment durationand the residual kidney potassium clearance are known. As in other massbalance models, the patient is assumed to be in steady state.

Steady State Mass Balance Model

The steady state mass balance model shown in FIG. 17 as relating tophosphorus can also be used to describe steady state potassium massbalance over a time averaged period for a patient treated byhemodialysis. The calculation of changes in serum potassiumconcentration, however, differs from the calculation of changes inphosphorus concentration described above. As set forth above, changes inthe serum potassium concentration during a treatment are represented bythe following equation,

$\begin{matrix}{\frac{({VC})}{t} = {{K_{M}\left( {C_{PRE} - C} \right)} - {K_{R}C} - {D\left( {C - C_{D}} \right)}}} & {E\text{-}{E1}}\end{matrix}$

wherein C is the serum or plasma concentration of potassium, C_(PRE) isthe predialysis serum concentration of potassium, C_(D) is the dialysatepotassium concentration, D is the dialyzer dialysance, and K_(R) is theresidual renal or kidney clearance. Equation E-E1 differs from the massbalance model for phosphorus by substituting dialysance for clearanceand allowing for a non-zero value for C_(D). As further set forth above,equation E-E1 can be solved to yield the following equation describingthe serum concentration of potassium during treatment.

$\begin{matrix}{\frac{C(t)}{C_{PRE}} = {\frac{K_{M} + {{DC}_{D}/C_{PRE}}}{K_{M} + K_{r} + D - Q_{UF}} + {\quad{\left\lbrack {1 - \frac{K_{M} + {{DC}_{D}/C_{PRE}}}{K_{M} + K_{r} + D - Q_{UF}}} \right\rbrack \times \left\lbrack \frac{V(t)}{V_{PRE}} \right\rbrack^{\frac{K_{M} + K_{r} + D - Q_{UF}}{Q_{UF}}}}}}} & {E\text{-}E\; 2}\end{matrix}$

The equation describing the change in serum potassium concentrationduring the interdialytic interval is the same as that of phosphorus.Based on equation E-D2, the equation describing the time-averagednormalized serum potassium concentration during treatment (n C _(tx))becomes

$\begin{matrix}{{n{\overset{\_}{C}}_{tx}} = {\frac{1}{t_{tx}}\left\{ {\left\lbrack \frac{\left( {K_{M} + {{DC}_{D}/C_{PRE}}} \right)t_{tx}}{K_{M} + K_{R} + D - Q_{UF}} \right\rbrack + {\quad{{\left\lbrack {1 - \frac{K_{M} + {{DC}_{D}/C_{PRE}}}{K_{M} + K_{R} + D - Q_{UF}}} \right\rbrack \left\lbrack \frac{V_{PRE}}{K_{M} + K_{R} + D} \right\rbrack}\left. \quad\left\lbrack {1 - \left( \frac{V_{POST}}{V_{PRE}} \right)^{\frac{({K_{M} + K_{R} + D})}{Q_{UF}}}} \right\rbrack \right\}}}} \right.}} & {E\text{-}E\; 3}\end{matrix}$

The overall mass balance relationship for potassium is then determinedby the following equation.

$\begin{matrix}{C_{PRE} = \frac{{G\left( {t_{tx} + \theta} \right)} + {D \times C_{D} \times t_{tx}}}{{\left( {D + K_{r}} \right) \times t_{tx} \times n{\overset{\_}{C}}_{tx}} + {K_{r} \times \theta \times n{\overset{\_}{C}}_{i}}}} & {E\text{-}E\; 4}\end{matrix}$

θ is the interdialytic interval, and n C _(i) is the normalized timeaveraged plasma potassium concentration for an interdialytic interval.Equation E-E4 is consistent with equations III-L and III-N above,wherein (t_(tx)+θ)=(10080/F). Equation E-E4 can be used to predict G ifthe pre-dialysis serum potassium concentration is measured in a patientwith knowledge of various treatment and patient parameters. Once G hasbeen calculated, it can be used in predicting the effect of changes inhemodialysis treatment parameters on the steady state serum potassiumconcentration using the equations set forth above. For example, once Gof the hemodialysis patient is known, C_(SS-PRE) of the patient underdifferent hemodialysis treatment conditions can be predicted byrearranging the equations described and utilizing the known G to solvefor C_(SS-PRE) of the hemodialysis patient. Using this model, the effectof specific therapy parameters (e.g., dialyzer potassium dialysances orclearances, weekly therapy frequency, therapy duration, etc.) onindividual hemodialysis patients' serum potassium levels can beevaluated.

An additional substantial difference between phosphorus and potassiummass balance in hemodialysis patients is the existence of substantialintestinal, specifically colonic, secretion of potassium. Normally, thesecretion of potassium by the intestines is approximately only 5% oftotal potassium elimination from the body, but it can range up to 35% ofdietary potassium in dialysis patients. The increase in secretion may bedue to an increase in the atypical potassium permeability of the largeintestinal epithelium. Independent of the mechanism discussed above, thesubstantial secretion of potassium by the intestines must be taken intoaccount in any mass balance model for potassium.

Methods

Two papers contain data that illustrates the effect of dialysatepotassium concentration on serum potassium levels and dialytic potassiumremoval. The first paper, Dolson et al., (Dolson et al., Acute decreasesin serum potassium augment blood pressure, Am J Kidney Dis 1995; 26:321-326), the entire contents of which are incorporated expressly hereinby reference and relied upon, describes the effect of differentdialysate potassium concentrations by studying eleven patients at 1, 2and 3 mEq/L dialysate potassium baths for at least one month each. Theresults from the study are shown in Table III.3.

TABLE III.3 Dialysate Potassium Level Predialysis Serum DialyticPotassium Removal (mEq/L) Potassium (mEq/L) (mEq) 1 4.9 ± 0.2 77.0 ± 6.52 5.1 ± 0.3 54.5 ± 7.9 3 5.3 ± 0.3 42.5 ± 9.9

The second paper, Zehnder et al., (Zehnder et al., Low-potassium andglucose-free dialysis maintains urea but enhances potassium removal,Nephrol Dial Transplant 2001; 16: 78-84), the entire contents of whichare incorporated expressly herein by reference and relied upon,describes the effect of 0, 1 or 2 mEq/L dialysate potassium baths forone week each. The results from the study are shown in Table III.4.

TABLE III.4 Dialysate Potassium Level Predialysis Serum DialyticPotassium Removal (mEq/L) Potassium (mEq/L) (mEq) 0 4.4 ± 0.2 117.1 ±10.3 1 4.5 ± 0.2 80.2 ± 6.2 2 4.9 ± 0.2 63.3 ± 5.2

FIG. 31 plots the data from Tables III.3 and III.4 together, where thedialytic potassium removal is normalized to that for a dialysatepotassium concentration of 0 mEq/L. The regression line has a negativeslope of 0.59 L/mEq. Assuming that the dietary potassium intake is equalin both groups, a decrease in serum potassium concentration by 1 mEq/Lresults in decreased secretion of potassium by the intestines by 59%because the total (intestinal plus dialytic) potassium removal mustalways be equal to the dietary potassium intake to maintain an overallbalance of potassium. In order to use this relationship in the massbalance model to increase or decrease the dialysate potassiumconcentration, the net generation rate needs to be modified based on theincrease or decrease of the serum concentration. Such a calculation isinterative in nature. Factoring this relationship to changes in serumpotassium concentration allows optimal prediction of dialysate potassiumconcentration to guide the decrease in the magnitude of the netgeneration of potassium (“G”) in equation E-E4 to account for the netsecretion of potassium. Otherwise, the mass balance model of potassiumfollows that previously described for phosphorus.

It should be understood that various changes and modifications to thepresently preferred embodiments described herein will be apparent tothose skilled in the art. Such changes and modifications can be madewithout departing from the spirit and scope of the present subjectmatter and without diminishing its intended advantages. It is thereforeintended that such changes and modifications be covered by the appendedclaims.

The invention is claimed as follows:
 1. A method of predicting serumpotassium concentrations in a patient during hemodialysis, the methodcomprising: measuring serum potassium concentrations (“C”) of thepatient over a hemodialysis treatment session time and anultrafiltration rate (“Q_(UF)”) calculated by a difference between pre-and post-dialytic body weight of the patient during an initialhemodialysis treatment session divided by a total treatment time of thetreatment session; estimating K_(M) and V_(PRE) for the patient using anon-linear least squares fitting to the equations: $\begin{matrix}{{C(t)} = {C_{PRE}\left\lbrack {{\frac{K_{M} + {{DC}_{D}/C_{PRE}}}{K_{M} + K_{R} + D - Q_{UF}} + \left. \quad{\left\lbrack {1 - \frac{K_{M} + {{DC}_{D}/C_{PRE}}}{K_{M} + K_{R} + D - Q_{UF}}} \right\rbrack \times \left\lbrack \frac{V(t)}{V_{PRE}} \right\rbrack^{\frac{K_{M} + K_{R} + D - Q_{UF}}{Q_{UF}}}} \right\rbrack},\mspace{20mu} {and}} \right.}} & \left( {{{eqn}.\mspace{14mu} 1}\text{-}A} \right) \\{\mspace{79mu} {{C(T)} = {C_{PRE} - {\left( {C_{PRE} - C_{POST}} \right)^{({- \frac{K_{M}T}{V_{PRE} - {Q_{UF}t_{tx}}}})}}}}} & \left( {{{eqn}.\mspace{14mu} 1}\text{-}B} \right)\end{matrix}$ wherein t is a time during the hemodialysis treatmentsession, T is a time after an end of the hemodialysis treatment session,t_(tx) is a total duration of the hemodialysis treatment session,C_(PRE) is a pre-dialysis plasma potassium concentration, C_(POST) is apost-dialytic plasma potassium concentration, C_(D) is a dialysatepotassium concentration, K_(M) is a potassium mobilization clearance ofthe patient, K_(R) is a residual renal clearance of potassium, D is adialyzer potassium dialysance, V_(PRE) is a pre-dialysis distributionvolume of potassium of the patient, andV(t)=V _(PRE) −Q _(UF) ×t  (eqn. 1-C); and predicting C of the patientat any time during any hemodialysis treatment session by using theequations 1-A and 1-B with the estimated K_(M) and V_(PRE) of thepatient.
 2. The method of claim 1, wherein K_(M) is determined using theequation: $\begin{matrix}{{K_{M} = {C_{POST}\left( \frac{K_{D} - Q_{UF}}{C_{PRE} - C_{POST}} \right)}},} & \left( {{{eqn}.\mspace{14mu} 1}\text{-}D} \right)\end{matrix}$ wherein K_(D) is a dialyzer potassium clearance estimatedusing the equation: $\begin{matrix}{K_{D} = {\frac{D}{2}{\left( {2 - \frac{C_{D}}{C_{PRE}} - \frac{C_{D}}{C_{POST}}} \right).}}} & \left( {{{eqn}.\mspace{14mu} 1}\text{-}E} \right)\end{matrix}$
 3. The method of claim 1, wherein D is determined usingthe equation: $\begin{matrix}{{D = {Q_{B}\frac{\left( {0.94 - {{Hct} \times 100}} \right)\left( {^{Z} - 1} \right)}{\left( {^{Z} - \frac{\left( {0.94 - {{Hct} \times 100}} \right)Q_{B}}{Q_{D}}} \right)}}},{wherein}} & \left( {{{eqn}.\mspace{14mu} 1}\text{-}F} \right) \\{{Z = {K_{O}A\frac{\left( {Q_{D} - {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B}}} \right)}{\left( {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B} \times Q_{D}} \right)}}},} & \left( {{{eqn}.\mspace{14mu} 1}\text{-}G} \right)\end{matrix}$ wherein Q_(B) is a blood flow rate, Q_(D) is a dialysateflow rate, K_(O)A is a dialyzer mass transfer area coefficient forpotassium, and Hct is a hematocrit count measured from patient's bloodsample.
 4. The method of claim 3, wherein D is calculated by assumingthat K_(O)A is 80% or about 80% of that for urea.
 5. The method of claim3, wherein K_(O)A is determined for using the equation: $\begin{matrix}{{{K_{O}A} = {\frac{\left( {0.94 - {{Hct} \times 100}} \right)Q_{B,M} \times Q_{D,M}}{Q_{D,M} - {\left( {0.94 - {{Hct} \times 100}} \right)Q_{BM}}} \times {\ln \left( \frac{1 - {D_{M}/Q_{D,M}}}{1 - {D_{M}/\left\lbrack {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B,M}} \right\rbrack}} \right)}}},} & \left( {{{eqn}.\mspace{14mu} 1}\text{-}H} \right)\end{matrix}$ and wherein Q_(B,M) is a previously measured blood flowrate that resulted in a measured dialyzer dialysance D_(M), and Q_(D,M)is a previously measured dialysate flow rate that resulted in themeasured dialyzer dialysance D_(M).
 6. The method of claim 1, wherein Cof the patient is measured every 15 minutes during the hemodialysistreatment session.
 7. The method of claim 1, wherein C of the patient ismeasured every 30 minutes during the hemodialysis treatment session. 8.The method of claim 1, wherein t_(tx) is 2 hours.
 9. The method of claim1, wherein t_(tx) is 4 hours.
 10. The method of claim 1, wherein t_(tx)is 8 hours.
 11. The method of claim 1, wherein T is 30 minutes.
 12. Themethod of claim 1, wherein T is 1 hour.
 13. The method of claim 1comprising determining V_(POST) using the equation:V _(POST) =V _(PRE) −Q _(UF) ×t _(tx)  (eqn. 1-I) and providing therapyto the patient based on the value of V_(POST).
 14. A method ofpredicting serum potassium concentrations in a patient duringhemodialysis, the method comprising: measuring serum potassiumconcentrations (“C”) of the patient over a hemodialysis treatmentsession time for an ultrafiltration rate (“Q_(UF)”)=0 during an initialhemodialysis treatment session; estimating K_(M) and V_(PRE) for thepatient using a non-linear least squares fitting to the equations:$\begin{matrix}{{C(t)} = {C_{PRE}\left\lbrack {\frac{K_{M} + {{DC}_{D}/C_{PRE}}}{K_{M} + K_{R} + D} + {\left. \quad{\left\lbrack {1 - \frac{K_{M} + {{DC}_{D}/C_{PRE}}}{K_{M} + K_{R} + D}} \right\rbrack ^{({{- t}\frac{K_{M} + K_{R} + D}{V_{PRE}}})}} \right\rbrack \mspace{20mu} {and}}} \right.}} & \left( {{{eqn}.\mspace{14mu} 2}\text{-}A} \right) \\{\mspace{79mu} {{C(T)} = {C_{PRE} - {\left( {C_{PRE} - C_{POST}} \right)^{({- \frac{K_{M}T}{V_{PRE}}})}}}}} & \left( {{{eqn}.\mspace{14mu} 2}\text{-}B} \right)\end{matrix}$ wherein t is a time during the hemodialysis treatmentsession, T is a time after an end of the hemodialysis treatment session,C_(PRE) is a pre-dialysis plasma potassium concentration, C_(POST) is apost-dialytic plasma potassium concentration, C_(D) is a dialysatepotassium concentration, K_(M) is a potassium mobilization clearance ofthe patient, K_(R) is a residual renal clearance of potassium, D is adialyzer potassium dialysance, and V_(PRE) is a pre-dialysisdistribution volume of potassium of the patient; and predicting C of thepatient at any time during any hemodialysis treatment session by usingthe equations 2-A and 2-B with the estimated K_(M) and V_(PRE) of thepatient.
 15. The method of claim 14, wherein K_(M) is determined usingthe equation: $\begin{matrix}{{K_{M} = {C_{POST}\left( \frac{K_{D} - Q_{UF}}{C_{PRE} - C_{POST}} \right)}},} & \left( {{{eqn}.\mspace{14mu} 2}\text{-}C} \right)\end{matrix}$ wherein K_(D) is a dialyzer potassium clearance estimatedusing the equation: $\begin{matrix}{K_{D} = {\frac{D}{2}{\left( {2 - \frac{C_{D}}{C_{PRE}} - \frac{C_{D}}{C_{POST}}} \right).}}} & \left( {{{eqn}.\mspace{14mu} 2}\text{-}D} \right)\end{matrix}$
 16. The method of claim 14, wherein D is determined usingthe equation: $\begin{matrix}{{D = {Q_{B}\frac{\left( {0.94 - {{Hct} \times 100}} \right)\left( {^{Z} - 1} \right)}{\left( {^{Z} - \frac{\left( {0.94 - {{Hct} \times 100}} \right)Q_{B}}{Q_{D}}} \right)}}},{wherein}} & \left( {{{eqn}.\mspace{14mu} 2}\text{-}E} \right) \\{{Z = {K_{O}A\frac{\left( {Q_{D} - {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B}}} \right)}{\left( {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B} \times Q_{D}} \right)}}},} & \left( {{{eqn}.\mspace{14mu} 2}\text{-}F} \right)\end{matrix}$ wherein Q_(B) is the blood flow rate, Q_(D) is thedialysate flow rate, K_(O)A is a dialyzer mass transfer area coefficientfor potassium, and Hct is a hematocrit count measured from patient'sblood sample.
 17. The method of claim 16, wherein D is determined usingthe equation: $\begin{matrix}{{D = {Q_{B}\frac{\left( {0.94 - {{Hct} \times 100}} \right)\left( {^{Z} - 1} \right)}{\left( {^{Z} - \frac{\left( {0.94 - {{Hct} \times 100}} \right)Q_{B}}{Q_{D}}} \right)}}},{wherein}} & \left( {{{eqn}.\mspace{14mu} 2}\text{-}G} \right) \\{{Z = {K_{O}A\frac{\left( {Q_{D} - {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B}}} \right)}{\left( {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B} \times Q_{D}} \right)}}},} & \left( {{{eqn}.\mspace{14mu} 2}\text{-}H} \right)\end{matrix}$ wherein Q_(B) is the blood flow rate, Q_(D) is thedialysate flow rate, K_(O)A is a dialyzer mass transfer area coefficientfor potassium, and Hct is hematocrit count measured from patient's bloodsample.
 18. The method of claim 16, wherein D is calculated by assumingthat K_(O)A is 80% or about 80% of that for urea.
 19. The method ofclaim 16, wherein K_(O)A is determined using the equation:$\begin{matrix}{{K_{o}A} = {\frac{\left( {0.94 - {{Hct} \times 100}} \right)Q_{B,M} \times Q_{D,M}}{Q_{D,M} - {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B,M}}} \times {\ln \left( \frac{1 - {D_{M}/Q_{D,M}}}{1 - {D_{M}/\left\lbrack {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B,M}} \right\rbrack}} \right)}}} & \left( {{{eqn}.\mspace{14mu} 2} - I} \right)\end{matrix}$ and wherein Q_(B,M) is a previously measured blood flowrate that resulted in a measured dialyzer dialysance D_(M), and Q_(D,M)is a previously measured dialysate flow rate that resulted in themeasured dialyzer dialysance D_(M).
 20. The method of claim 14, whereinC of the patient is measured every 15 minutes during the hemodialysistreatment session.
 21. The method of claim 14, wherein C of the patientis measured every 30 minutes during the hemodialysis treatment session.22. The method of claim 14, wherein the total duration of thehemodialysis treatment session is 2 hours.
 23. The method of claim 14,wherein the total duration of the hemodialysis treatment session is 4hours.
 24. The method of claim 14, wherein the total duration of thehemodialysis treatment session is 8 hours.
 25. The method of claim 14,wherein T is 30 minutes.
 26. The method of claim 14, wherein T is 1hour.
 27. The method of claim 14 comprising determining V_(POST) usingthe equation:V _(POST) =V _(PRE)  (eqn. 2-J) and providing therapy to the patientbased on the value of V_(POST).
 28. A computing device comprising: adisplay device; an input device; a processor; and a memory device thatstores a plurality of instructions, which when executed by theprocessor, cause the processor to operate with the display device andthe input device to: (a) receive data relating to serum potassiumconcentrations (“C”) of a hemodialysis patient over a hemodialysistreatment session time and an ultrafiltration rate (“Q_(UF)”) calculatedby a difference between pre- and post-dialytic body weight of thehemodialysis patient during a hemodialysis treatment session divided bya total treatment time of the treatment session; (b) estimate K_(M) andV_(PRE) for the patient using a non-linear least squares fitting to theequations: $\begin{matrix}{{{C(t)} = {C_{PRE}\begin{bmatrix}{\frac{K_{M} + {{DC}_{D}/C_{PRE}}}{K_{M} + K_{R} + D - Q_{UF}} +} \\{\left\lbrack {1 - \frac{K_{M} + {{DC}_{D}/C_{PRE}}}{K_{M} + K_{R} + D - Q_{UF}}} \right\rbrack \times} \\\left\lbrack \frac{V(t)}{V_{PRE}} \right\rbrack^{\frac{K_{M} + K_{R} + D - Q_{UF}}{Q_{UF}}}\end{bmatrix}}}{and}} & \left( {{{eqn}.\mspace{14mu} 3} - A} \right) \\{{C(T)} = {C_{PRE} - {\left( {C_{PRE} - C_{POST}} \right)e^{({- \frac{K_{M}T}{V_{PRE} - {Q_{UF}t_{tx}}}})}}}} & \left( {{{eqn}.\mspace{14mu} 3} - b} \right)\end{matrix}$ wherein t is a time during the hemodialysis treatmentsession, T is a time after an end of the hemodialysis treatment session,t_(tx) is a total duration of the hemodialysis treatment session,C_(PRE) is a pre-dialysis plasma potassium concentration, C_(POST) is apost-dialytic plasma potassium concentration, C_(D) is a dialysatepotassium concentration, K_(M) is a potassium mobilization clearance ofthe patient, K_(R) is a residual renal clearance of potassium, D is adialyzer potassium dialysance, V_(PRE) is a pre-dialysis distributionvolume of potassium of the hemodialysis patient, and (c) predict C ofthe patient at any time during hemodialysis by using the equation 3-Aand the estimated K_(M) and V_(PRE) of the patient.
 29. The computingdevice of claim 28, wherein if Q_(UF)=0, then $\begin{matrix}{{{C(t)} = {C_{PRE}\begin{bmatrix}{\frac{K_{M} + {{DC}_{D}/C_{PRE}}}{K_{M} + K_{R} + D} +} \\{\left\lbrack {1 - \frac{K_{M} + {{DC}_{D}/C_{PRE}}}{K_{M} + K_{R} + D}} \right\rbrack e^{({{- t}\frac{K_{M} + K_{R} + D}{V_{PRE}}})}}\end{bmatrix}}}\mspace{20mu} {and}} & \left( {{{eqn}.\mspace{14mu} 3} - C} \right) \\{\mspace{79mu} {{C(T)} = {C_{PRE} - {\left( {C_{PRE} - C_{POST}} \right){e^{({- \frac{K_{M}T}{V_{PRE}}})}.}}}}} & \left( {{{eqn}.\mspace{14mu} 3} - D} \right)\end{matrix}$
 30. The computing device of claim 28, wherein theprocessor operates with the display device and the input device toreceive data relating to at least one of K_(R), D or a sampling time forcollecting the serum potassium concentration.
 31. A method ofdetermining steady state, pre-dialysis serum potassium levels in ahemodialysis patient, the method comprising: obtaining a net generationof potassium (“G”) of the hemodialysis patient_(;) determining steadystate, pre-dialysis serum potassium levels (“C_(SS-PRE)”) of thehemodialysis patient using the equation: $\begin{matrix}{C_{{SS} - {PRE}} = \frac{{G\left( {10080/F} \right)} + {D \times C_{D}t_{tx}}}{{\left( {D + K_{R}} \right)n{\overset{\_}{C}}_{tx}t_{tx}} + {K_{R}n{{\overset{\_}{C}}_{i}\left( {{10080/F} - t_{tx}} \right)}}}} & \left( {{{eqn}.\mspace{14mu} 4} - A} \right)\end{matrix}$ wherein C_(D) is a dialysate potassium concentration,K_(R) is a residual renal clearance of potassium, t_(tx) a totalduration of the hemodialysis treatment session, D is a dialyzerpotassium dialysance, F is a frequency of treatments per week, n C _(tx)is a normalized time averaged plasma potassium concentration during adialysis treatment, and n C _(i) is a normalized time averaged plasmapotassium concentration for an interdialytic interval; and simulatingthe effect of at least one of a patient parameter or a treatmentparameter on C_(SS-PRE) of the hemodialysis patient.
 32. The method ofclaim 31, wherein n C _(tx) and n C _(i) are determined using theequations: $\begin{matrix}{{{n{\overset{\_}{C}}_{tx}} = {\frac{1}{tx}\begin{Bmatrix}{\left\lbrack \frac{\left( {K_{M} + {{DC}_{D}/C_{PRE}}} \right)t_{tx}}{K_{M} + K_{R} + D - Q_{UF}} \right\rbrack +} \\{{\left\lbrack {1 - \frac{K_{M} + {{DC}_{D}/C_{PRE}}}{K_{M} + K_{R} + D - Q_{UF}}} \right\rbrack \left\lbrack \frac{V_{PRE}}{K_{M} + K_{R} + D} \right\rbrack}\left\lbrack {1 - \left( \frac{V_{POST}}{V_{PRE}} \right)^{\frac{({K_{M} + K_{R} + D})}{Q_{UF}}}} \right\rbrack}\end{Bmatrix}}},{and}} & \left( {{{eqn}.\mspace{14mu} 4} - B} \right) \\{{{n{\overset{\_}{C}}_{i}} = {\frac{1}{{10080/F} - t_{tx}}\left\{ {\begin{matrix}{\left\lbrack \frac{K_{M}\left( {{1000/F} - t_{tx}} \right)}{K_{M} + K_{R} + Q_{WG}} \right\rbrack +} \\\begin{bmatrix}{\frac{K_{M} + {{DC}_{D}/C_{PRE}}}{K_{M} + K_{R} + D - Q_{UF}} +} \\{{\left\lbrack {1 - \frac{K_{M} + {D\; {C_{D}/C_{PRE}}}}{K_{M} + K_{R} + D - Q_{UF}}} \right\rbrack \left\lbrack \frac{V_{POST}}{V_{PRE}} \right\rbrack}^{\frac{({K_{M} + K_{R} + D})}{Q_{UF}}} -} \\\frac{K_{M}}{K_{M} + K_{R} + Q_{WG}}\end{bmatrix} \\{\left\lbrack \frac{V_{POST}}{K_{M} + K_{R}} \right\rbrack \left\lbrack {1 - \left( \frac{V_{POST}}{V_{PRE}} \right)^{\frac{({K_{M} + K_{R}})}{Q_{WG}}}} \right\rbrack}\end{matrix} \times} \right\}}},} & \left( {{{eqn}.\mspace{14mu} 4} - C} \right)\end{matrix}$ wherein K_(M) is a potassium mobilization clearance of thepatient, Q_(WG) is a constant rate of fluid gain by the patient duringthe interdialytic time interval, C_(PRE) is the pre-dialysis serumpotassium concentration, Q_(UF) is a constant rate of fluid removed fromthe patient, V_(PRE) is a pre-dialysis distribution volume of potassiumof the patient prior to a hemodialysis treatment session, and V_(POST)is a post-dialysis value of potassium of the patient at the end of ahemodialysis treatment session.
 33. The method of claim 31, whereinthere is negligible net ultrafiltration or fluid removal from thepatient during hemodialysis therapies and no weight gain betweenhemodialysis therapies, Q_(UF)=0, and wherein n C _(tx) and n C _(i) aredetermined using the equations: $\begin{matrix}{{{n{\overset{\_}{C}}_{tx}} = {\frac{1}{tx}\begin{Bmatrix}{\left\lbrack \frac{\left( {K_{M} + {{DC}_{D}/C_{PRE}}} \right)t_{tx}}{K_{M} + K_{R} + D} \right\rbrack +} \\{{\left\lbrack {1 - \frac{K_{M} + {{DC}_{D}/C_{PRE}}}{K_{M} + K_{R} + D}} \right\rbrack \left\lbrack \frac{V_{PRE}}{K_{M} + K_{R} + D} \right\rbrack}\left\lbrack {1 - e^{(\frac{{- {({D + K_{R} + K_{M}})}}t_{tx}}{V_{PRE}})}} \right\rbrack}\end{Bmatrix}}},{and}} & \left( {{{eqn}.\mspace{14mu} 4} - D} \right) \\{{{n{\overset{\_}{C}}_{i}} = {\frac{1}{{10080/F} - t_{tx}}\left\{ {\begin{matrix}{\left\lbrack \frac{K_{M}\left( {{1000/F} - t_{tx}} \right)}{K_{M} + K_{R}} \right\rbrack +} \\\begin{bmatrix}{\frac{K_{M} + {{DC}_{D}/C_{PRE}}}{K_{M} + K_{R} + D} +} \\\begin{matrix}{\left\lbrack {1 - \frac{K_{M} + {D\; {C_{D}/C_{PRE}}}}{K_{M} + K_{R} + D}} \right\rbrack \times} \\e^{\;^{(\frac{{- {({D + K_{R} + K_{M}})}}t_{tx}}{V_{PRE}})} -}\end{matrix} \\\frac{K_{M}}{K_{M} + K_{R}}\end{bmatrix} \\{\left\lbrack \frac{V_{POST}}{K_{M} + K_{R}} \right\rbrack \left\lbrack {1 - e^{(\frac{{- {({K_{R} + K_{M}})}}{({{10080/F} - t_{tx}})}}{V_{POST}})}} \right\rbrack}\end{matrix} \times} \right\}}},} & \left( {{{eqn}.\mspace{14mu} 4} - E} \right)\end{matrix}$ wherein K_(M) is a potassium mobilization clearance of thepatient, Q_(WG) is a constant rate of fluid gain by the patient duringthe interdialytic time interval, C_(PRE) is the pre-dialysis serumpotassium concentration Q_(UF) is a constant rate of fluid removed fromthe patient, V_(PRE) is a pre-dialysis distribution volume of potassiumof the patient prior to a hemodialysis treatment session, and V_(POST)is a post-dialysis value of potassium of the patient at the end of ahemodialysis treatment session.
 34. The method of claim 32, wherein thepatient parameter is G, K_(M) or V_(PRE).
 35. The method of claim 31,wherein the treatment parameter is t_(tx), D or F.
 36. The method ofclaim 32, wherein K_(M) is determined using the equation:$\begin{matrix}{{K_{M} = {C_{POST}\left( \frac{K_{D} - Q_{UF}}{C_{PRE} - C_{POST}} \right)}},} & \left( {{{eqn}.\mspace{14mu} 4} - F} \right)\end{matrix}$ wherein K_(D) is a dialyzer potassium clearance estimatedusing the equation: $\begin{matrix}{{K_{D} = {\frac{D}{2}\left( {2 - \frac{C_{D}}{C_{PRE}} - \frac{C_{D}}{C_{POST}}} \right)}},} & \left( {{{eqn}.\mspace{14mu} 4} - G} \right)\end{matrix}$ and C_(POST) is a post-dialytic plasma potassiumconcentration, and C_(PRE) is a pre-dialysis plasma potassiumconcentration.
 37. The method of claim 31, wherein G is calculated usingthe equation: $\begin{matrix}{G = {\left\lbrack \frac{{\left( {D + K_{R}} \right)n{\overset{\_}{C}}_{tx}t_{tx}} + {K_{R}n{{\overset{\_}{C}}_{i}\left( {{10080/F} - t_{tx}} \right)}}}{\left( {10080/F} \right)} \right\rbrack \times {\quad\begin{bmatrix}{C_{{SS} - {PRE} - {IN}} -} \\\frac{D \times C_{D} \times t_{tx}}{{\left( {D + K_{R}} \right)n{\overset{\_}{C}}_{tx}t_{tx}} + {K_{R}n{{\overset{\_}{C}}_{i}\left( {{10080/F} - t_{tx}} \right)}}}\end{bmatrix}}}} & \left( {{{eqn}.\mspace{14mu} 4} - H} \right)\end{matrix}$ wherein C_(SS-PRE-IN) is an initial, measured, steadystate, pre-dialysis serum potassium level of the hemodialysis patientwho is maintained by a hemodialysis therapy for a specified time priorto the calculation of G using equation 4-G.
 38. The method of claim 31,wherein D is determined using the equation: $\begin{matrix}{{D = {Q_{B}\frac{\left( {0.94 - {{Hct} \times 100}} \right)\left( {e^{Z} - 1} \right)}{\left( {e^{Z} - \frac{\left( {0.94 - {{Hct} \times 100}} \right)Q_{B}}{Q_{D}}} \right)}}},{wherein}} & \left( {{{eqn}.\mspace{14mu} 4} - I} \right) \\{{Z = {K_{O}A\frac{\left( {Q_{D} - {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B}}} \right)}{\left( {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B} \times Q_{D}} \right)}}},} & \left( {{{eqn}.\mspace{14mu} 4} - J} \right)\end{matrix}$ wherein Q_(B) is the blood flow rate, Q_(D) is thedialysate flow rate, K_(O)A is a dialyzer mass transfer area coefficientfor potassium, and Hct is a hematocrit count measured from patient'sblood sample.
 39. The method of claim 38, wherein D is calculated byassuming that K_(O)A is 80% or about 80% of that for urea.
 40. Themethod of claim 38, wherein K_(O)A is determined using the equation:$\begin{matrix}{{{K_{O}A} = {\frac{\left( {0.94 - {{Hct} \times 100}} \right)Q_{B,M} \times Q_{D,M}}{Q_{D,M} - {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B,M}}} \times {\ln \left( \frac{1 - {D_{M}/Q_{D,M}}}{1 - {D_{M}/\left\lbrack {\left( {0.94 - {{Hct} \times 100}} \right)Q_{B,M}} \right\rbrack}} \right)}}},} & \left( {{{eqn}.\mspace{14mu} 4} - K} \right)\end{matrix}$ and wherein Q_(B,M) is a previously measured blood flowrate that resulted in a measured dialyzer dialysance D_(M), and Q_(D,M)is a previously measured dialysate flow rate that resulted in a measureddialyzer dialysance D_(M).
 41. The method of claim 31 comprisingdetermining a level of potassium intake so that C_(SS-PRE) of thehemodialysis patient ranges between about 3.5 mEq/L and 5.3 mEq/L. 42.The method of claim 31 comprising determining a potassium binderadministered to the patient so that C_(SS-PRE) of the hemodialysispatient ranges between about 3.5 mEq/L and 5.3 mEq/L.
 43. The method ofclaim 31 comprising determining a total hemodialysis treatment sessiontime so that C_(SS-PRE) of the hemodialysis patient ranges between about3.5 mEq/L and 5.3 mEq/L.
 44. The method of claim 31 comprisingdetermining the frequency F so that C_(SS-PRE) of the hemodialysispatient ranges between about 3.5 mEq/L and 5.3 mEq/L.
 45. The method ofclaim 31 comprising determining a required blood flowrate and/or adialysate flowrate so that C_(SS-PRE) of the hemodialysis patient rangesbetween about 3.5 mEq/L and 5.3 mEq/L.
 46. The method of claim 31comprising determining an amount of potassium added to the dialysate sothat C_(SS-PRE) of the hemodialysis patient ranges between about 3.5mEq/L and 5.3 mEq/L.
 47. The method of claim 31, wherein K_(M) andV_(PRE) are determined by measuring serum potassium concentrations (“C”)of the hemodialysis patient over a hemodialysis treatment session timeand an ultrafiltration rate (“Q_(UF)”) calculated by a differencebetween pre- and post-dialytic body weight of the patient during aninitial hemodialysis treatment session divided by a total treatment timeof the treatment session, and calculating K_(M) and V_(PRE) for thehemodialysis patient using a non-linear least squares fitting to theequations: $\begin{matrix}{{{C(t)} = {C_{PRE}\begin{bmatrix}{\frac{K_{M} + {{DC}_{D}/C_{PRE}}}{K_{M} + K_{R} + D - Q_{UF}} +} \\{\left\lbrack {1 - \frac{K_{M} + {{DC}_{D}/C_{PRE}}}{K_{M} + K_{R} + D - Q_{UF}}} \right\rbrack \times} \\\left\lbrack \frac{V(t)}{V_{PRE}} \right\rbrack^{\frac{K_{M} + K_{R} + D - Q_{UF}}{Q_{UF}}}\end{bmatrix}}}\mspace{20mu} {and}} & \left( {{{eqn}.\mspace{14mu} 4} - L} \right) \\{\mspace{79mu} {{C(T)} = {C_{PRE} - {\left( {C_{PRE} - C_{POST}} \right)e^{({- \frac{K_{M}T}{V_{PRE} - {Q_{UF}t_{tx}}}})}}}}} & \left( {{{eqn}.\mspace{14mu} 4} - M} \right)\end{matrix}$ wherein t is a time during the hemodialysis treatmentsession, T is a time after an end of the hemodialysis treatment session,C_(PRE) is a pre-dialysis plasma potassium concentration, C_(POST) is apost-dialytic plasma potassium concentration, C_(D) is a dialysatepotassium concentration, K_(M) is a potassium mobilization clearance ofthe patient, K_(R) is a residual renal clearance of potassium, V_(PRE)is a pre-dialysis distribution volume of potassium of the patient, andV(t)=V _(PRE) −Q _(UF) ×t  (eqn. 4-N).
 48. A computing devicecomprising: a display device; an input device; a processor; and a memorydevice that stores a plurality of instructions, which when executed bythe processor, cause the processor to operate with the display deviceand the input device to: (a) receive data relating to a net generationof potassium (“G”) from at least a dietary potassium intake of ahemodialysis patient or a urea kinetic modeling of the hemodialysispatient; (b) determine steady state, pre-dialysis serum potassium levels(“C_(SS-PRE)”) of the patient using the equation: $\begin{matrix}{C_{{SS} - {PRE}} = \frac{{G\left( {10080/F} \right)} + {D \times C_{D} \times t_{tx}}}{{\left( {D + K_{R}} \right)n{\overset{\_}{C}}_{tx}t_{tx}} + {K_{R}n{{\overset{\_}{C}}_{i}\left( {{10080/F} - t_{tx}} \right)}}}} & \left( {{{eqn}.\mspace{14mu} 5} - A} \right)\end{matrix}$ wherein C_(D) is a dialysate potassium concentration,K_(R) is a residual renal clearance of potassium, t_(tx) a totalduration of the hemodialysis treatment session, D is a dialyzerpotassium dialysance, F is a frequency of treatments per week, n C _(tx)is a normalized time averaged plasma potassium concentration during adialysis treatment, and n C _(i) is a normalized time averaged plasmapotassium concentration for an interdialytic interval; and (c) simulatethe effect of at least one of a patient parameter or a treatmentparameter on the C_(SS-PRE) of the hemodialysis patient.
 49. Thecomputing device of claim 48, wherein the processor operates with thedisplay device and the input device to receive data relating to at leastone of C_(D), D, K_(R), K_(M), V_(PRE), t_(tx), F and C_(PRE) about amonth before a hemodialysis treatment session or a sampling time forcollecting the serum potassium concentration, wherein K_(M) is apotassium mobilization clearance of the patient, V_(PRE) is apre-dialysis distribution volume of potassium of the patient, andC_(PRE) is a pre-dialysis plasma potassium concentration.
 50. Thecomputing device of claim 48, wherein the computing device displays atreatment regimen of the hemodialysis patient so that C_(SS-PRE) iswithin a desired range.
 51. A computing device comprising: a displaydevice; an input device; a processor; and a memory device that stores aplurality of instructions, which when executed by the processor, causethe processor to operate with the display device and the input deviceto: (a) determine a net generation of potassium (“G”) using theequation: $\begin{matrix}{G = {\left\lbrack \frac{{\left( {D + K_{R}} \right)n{\overset{\_}{C}}_{tx}t_{tx}} + {K_{R}n{{\overset{\_}{C}}_{i}\left( {{10080/F} - t_{tx}} \right)}}}{\left( {10080/F} \right)} \right\rbrack \times {\quad\begin{bmatrix}{C_{{SS} - {PRE} - {IN}} -} \\\frac{D \times C_{D} \times t_{tx}}{{\left( {D + K_{R}} \right)n{\overset{\_}{C}}_{tx}t_{tx}} + {K_{R}n{{\overset{\_}{C}}_{i}\left( {{10080/F} - t_{tx}} \right)}}}\end{bmatrix}}}} & \left( {{{eqn}.\mspace{14mu} 6} - A} \right)\end{matrix}$ wherein C_(SS-PRE-IN) is an initial, measured, steadystate, pre-dialysis serum potassium level of the hemodialysis patientwho is maintained by a hemodialysis therapy for a specified time priorto the calculation of G using equation 6-A, C_(D) is a dialysatepotassium concentration, K_(R) is a residual renal clearance ofpotassium, t_(tx) a total duration of the hemodialysis treatmentsession, D is a dialyzer potassium dialysance, F is a frequency oftreatments per week, n C _(tx) is a normalized time averaged plasmapotassium concentration during a dialysis treatment, and n C _(i) is anormalized time averaged plasma potassium concentration for aninterdialytic interval; (b) predict steady state, pre-dialysis serumpotassium levels (“C_(SS-PRE)”) of the hemodialysis patient using theequation: $\begin{matrix}{{C_{{SS} - {PRE}} = \frac{{G\left( {10080/F} \right)} + {D \times C_{D} \times t_{tx}}}{{\left( {D + K_{R}} \right)n{\overset{\_}{C}}_{tx}t_{tx}} + {K_{R}n{{\overset{\_}{C}}_{i}\left( {{10080/F} - t_{tx}} \right)}}}};} & \left( {{{eqn}.\mspace{14mu} 6} - B} \right)\end{matrix}$  and (c) simulate the effect of at least one of a patientparameter or a treatment parameter on C_(SS-PRE) of the hemodialysispatient.
 52. The computing device of claim 51, wherein the processoroperates with the display device and the input device to receive datarelating to at least one of C_(D), D, K_(R), K_(M), V_(PRE), t_(tx), Fand C_(PRE) about a month before a hemodialysis treatment session or asampling time for collecting the serum potassium concentration, whereinK_(M) is a potassium mobilization clearance of the patient, V_(PRE) is apre-dialysis distribution volume of potassium of the patient, andC_(PRE) is a pre-dialysis plasma potassium concentration.
 53. Thecomputing device of claim 51, wherein the computing device displays atreatment regimen of the hemodialysis patient so that C_(SS-PRE) iswithin a desired range.